THE MATHEMATICAL THEORY OF
PROBABILITIES
■y^y^-
THE MACMILLAN COMPANY
NEW YORK BOSTON CHICAGO
DALLAS SAN FRANCISCO
MACMILLAN & CO.. Limited
LONDON BOMBAY CALCUTTA
MELBOURNE
THE MACMILLAN CO. OF CANADA. Ltd.
TORONTO
THE MATHEMATICAL THEORY
OF
PROBABILITIES
AND ITS Al'PLKWTIOX TO
FREQUENCY CURVES AND STATISTICAL
METHODS
ARNE FISHER.
TRANSLATED FROM THE DANISH
BY
rilARLOTTE DICKSON, B.A.
(COLUMBIA)
MATHEMATieAL ASSISTANT IN THE DEPARTMENT OF DEVELOPMENT
AND HK8EAKCH OF
THE AMERICAN TELEPHONE AND TELEGRAPH COMPANY
AND
\villia:\i bonynge, b.a.
(BELFAST)
WITH INTRODUCTORY NOTES
BY
M. C. RORTY
AND
F. W. FRANKLAND, F.I.A., F.A.S, F.S.S.
VOLUME I
Mathematical Probabilities, P'requency Curves, Homogradk and
Heterograde Statistics
SECOND EDITION GREATLY ENLARGED
NEW YORK
THE MACMILLAN COMPANY
1922
Copyright, 1915 and 1922,
By ARNE fisher
Set up and electrotj-ped. Published November, 1915.
Second Edition, greatly enlarged, May, 1922.
r*IN1KI> IN THE IT.SlTEll STATES Of AMEBICA
y^^ UNIVERSITY OF CALIFORNIA
^ 7_3 SANTA BARBARA COLLEGE LIBRARY
/9J^^ 69545
V, f
INTRODUCTORY NOTE TO THE SECOND EDITION.
Mr. Fisher has requested that an introduction be written to
this, the second edition of his work on probabilities, which shall
indicate some of the practical applications of the mathematical
theory with which his treatise deals.
The writer has only a limited knowledge of mathematical
technique — yet it has so happened that in twenty-five years of
active work as engineer, statistician and executive he has had
frequent occasion to call upon the skill of trained mathematicians
for the solution of practical problems involving frequency curves
and pi-obal)ilities. Among such mathematicians none has been
more helpful, or quicker to perceive the possibility of making
valuable applications of higher mathematics to business problems,
than ]\Ir. Fisher himself. For this reason it is a duty as well as a
privilege to outline, at his request, certain actual practical expe-
riences with mathematical applications and to indicate such
possible applications for the future.
The writer's initial experience with frequency curves and
probabilities was in the j-ears 1902 and 1903, when it became
evident, in analyzing various problems in telephone traffic, that
certain peak loads, which were superimposed upon the normal
seasonal, weekly, and daily fluctuations, could be accounted for
only by the laws of chance. Recourse was, therefore, had to the
formulie then available for approximate summations of the terms
of the binomial expansion, and from these a series of curves was
drawn which indicated for an}' given normal hourly traffic (as
indicated by studies of seasonal, weekly, and daily variations) the
probability that any given short period load would be equalled or
exceeded. Practical experience with these curves soon showed that,
in spite of minor errors, they were close enough to the real facts
to make them of primary importance in traffic studies of all kinds,
and particularly in the development of mechanical switching de-
vices. Their use for such purposes has now become a conmion-
place in telephone engineering.
As a by-product of the preceding application there have been
other interesting uses of the same probability curves. Effective
studies have been made of the decrease in the total stocks of small
machine parts that could be made possible by standardizing and
VI INTRODUCTORY NOTE TO THE SECOND EDITION.
reducing the number of types of screws, bolts, nuts, etc. The
curves can also be applicnl directly to every line of business and
every type of operation where prompt service must be given and
where the demand arises from a large number of independent
sources, and is, therefore, subject to p(^ak loads determined by the
laws of chance, which may be superimposed upon other "normal"
peak loads varying with the days of the week, the hours of the
day, etc.
Entirely separate applications of frequency curves are those
necessary in actuarial work. T'hese are relatively well known.
But it is less generally known that one of the most important of
business problems, that of depreciation, can be treated (effectively
only when approached on an actuarial basis with a full under-
standing of the fi-equency curves which govern the displacement,
year b}^ year, of the physical units involved.
A still further use of frequency curves and the theory of probabil-
ities, which is of immediate practical importance, is in connection
with sampling operations. The theory of sampling has already
been well developed, but adequate efforts have not yet been made
by mathematicians to reduce the processes of sampling to de-
pendable simple rules that can be applied by business executives
and statisticians untrained in higher mathematics. In census
work, and in statistical and other reports made by business or-
ganizations, the waste of money, that could be avoided by an
intelligent application of the theory of sampling, is very great.
Not only can many reports and analyses be made much more
cheaply and quickly by sampling processes, but they can also be
made more accurately. Many important items of information can
be determined only by trained specialists. In such cases the only
procedure, that does not involve prohibitive expense in large census
operations, is to tie such items, by a sampling process, to other
items which ai-e susceptible of exact enumeration by relatively
unskilled enumerators, and then to compute the totals for the
special items from the relations of such items to the items which
are completely enumerated.
All of the preceding are in the field of inmiediate practicalities.
When we come to the future, one of the most promising uses of
mathematics is in the d(>velopment of logical processes. It is not
going too far to say that all business, and most engineering opera-
tions are fundamentally based on probabilities. The business man
is always dealing in degrees of uncertainty, and even the engineer
INTRODUCTORY NOTE TO THE SECOND EDITION. VU
has only occasionally a definite set of conditions ujion which to
base his computations. Where the problem is primarily- a financial
one, he must balance the cost of overbuilding agamst the cost of
underbuilding; and, if he combines business judgment with en-
gineering skill, he will multiply the amount of each possible loss
by the probability of its occurring, and will ordinarily choose,
among all possible plans, the plan which involves the minimum
probable loss. Here it is not inappropriate to interject the idea
that the most practical logic must always be in terms of probabil-
ities, and that a logic which deals, or pretends to deal, in certainties
only is not alone useless, but is also harmful and misleading, when
difficult problems are to be approached. Such problems can
rarely, if ever, be solved except through the cumulation toward a
certainty of many small probabilities established from un corre-
lated, or only partialh-^ correlated, viewpoints.
A final suggestion which is to-day speculative, but may assume
important practical aspects in the near future, is with respect to
the applications of frequency curves and probabilities to phj-sical
and cosmic mathematics. In such mathematics we are forced to
assume that all of our measures must arise out of the things meas-
ured. When we deal with physical velocities, it would seem that
our only measures of velocity can arise out of the velocities them-
selves. Similar considerations hold true with respect to funda-
mental measures of physical extension. Under these circum-
stances we may talk in terms of infinite space and of infinite time,
but we can hardly talk in terms of infinites when we are dealing
with the dimensions of atomic structure and the velocities of
material particles. In these cases it seems very highly probable
that we are dealing with frequency distributions which we must
measure and define in terms derived from such distributions them-
selves. With respect to such measures some of our frequency
curves may have hifinite "tails," but it is more proljable that the
frequency forms are such that they can be completely defined in
finite terms. Along this same line, we may even risk a closing
speculation that the relative proportions of organized matter and
space in the stellar universe are determined through the opera-
tions of the laws of chance in establishing heterogeneities in what
is otherwise a homogeneous void-filling medium.
M. C. RORTY.
New York City,
March 2, 1922.
PREFACE TO THE SECOND EDITION.
At the time when the first edition of this little book was published
in 1916, I ex})ected to issue a second volume shortly after, dealing
with fre(|uency curves and frequency surfaces as well as the re-
lated problem of co-variation (correlation). The manuscript for
this volume was completed and i^rinting had already connnenced
on some of the; chapters, when a series of misfortunes, not neces-
sarily unexpected, oven-took the work. A major part of the manu-
script while in transit to a friend in Denmark for review and cor-
rections went down with a Danish vessel when torpedoed by an
outlaw German submarine. A duplicate copy was for some i-eason
or other withhc^ld by the British rnilitarj^ censor and not i-eturned
to the writer until lonp; after the termination of the world war.
My third and final copy of the manuscript, which I had submitted
to an American friend for critical review was also lost in transit.
The veritable nemesis which seems to have followed my efforts is,
however, only a verification of the all prevailing laws of chance,
which ever}^ serious minded student must face with unperturbed
attitude. In fact, the al)ove misfortunes have, after all, only
made me more determined to complete another collection of notes,
which I eventually hope to put into proper shape for publication.
In the meantime the first edition has been out of print for more
than two years, and when the publisher asked me to prepare a
new edition I took advantage of this opportunity to add several
chapters on frequency functions and their application to het-
erograde statistical series so as to give a complete treatment of
statistical functions involving one variable. The book is, there-
fore, twice its original size and contains the major part of what I
originally intended for a second volume.
The reader will readily notice that my treatment of the subject
is based throughout upon the principles of the classical prol^ability
theory as founded by Bernoulli, De JMoivre and above all by the
great Laplace and his disciple, Poisson, I am of the opinion that
these principles and their further extension by the Scandinavian
statisticians and actuaries. Gram, Thiele, Westergaard, Charlier,
Wicksell and Jorgensen, offer as yet the best and also the most
powerful tools for the treatment of collected statistical data by
means of mathematical methods. In the way of admnbration and
X PREFACE TO THE SECOXI) KDITIOX.
economy of thouiilit the Laplaccan methods stand unsurpassed
in the whole reahn of math(Miiatical statistics. I have, therefore,
in this volume limited nn' investi<>;ations to a systematic treat-
ment along these lines. I hope, however, in the forthcoming
second volunu^ to treat the methods of Pearson, Edgeworth,
Kapteyn, Bachelier and Knibbs and show tlunr inflation to La-
place's theory.
Th(» reason why the Laplacean doctrine^ of freciuency curves lias
l)een ignored until comixiratively recent years and has remained
more or h^ss obscure is })erhaps tlue to the fact that for moi-e than
a cent my it remained a theory Jiure and simple and was used l)ut
sparingly in pi-actical calculations.
Any statistical theory, in order to be of use in practical work,
must be arranged in such a manner that it is readily adaptable to
numerical compulations. Advanced mathematical computation
has not been given its due reward and proper att(^ntion in our
ordinaiy academic instruction. A high grade mathematical
computer is indeed a ''rare bird," nmch more so in fact than a
good mathematician. To arrange and plan the numerical work
in connection with the theoretical fonnulae so that the detailed and
painstaking work is reduced to a mininmm, and at the same time
afford the pi'oper means for checking and counterclu^cking, is by
no means an easy task and often requires as much ingenuity as
the actual development of the theoretical fornmhip. While Gauss
has always been acknowledged as one of the world's gr(^at(»st com-
puters and in addition to his extensive work in pure mathematics
also did much practical work in surveying, physics, and in financial
and actuarial investigations, Laplace during his entire career
remained a pure mathematician and apparently failed to grasp the
paramount attributes required by a successful (computer. His
attempt to inject himself into pul)lic life, as for instance when he
secured for himself an a])pointment as minister of the interior,
must be regarded as a dismal failure as admitttnl in Najjoleon's
memorandum on his dismissal.
The failure of Laplace to recognize fully the all-important phase
of numerical computations in all observations on statistical mass
phenomena is in my opinion the main reason why the Gaussian
theoi\- of observations and the allied subject on the theory of least
squai-es has hitherto supj^lanted the admittedly superior theory
of the great Frenchman. Gauss in addition to his theory furnished
an essentially useful and elegant method for performing the nec(>s-
PREFACE TO THE SECOND EDITION. XI
sary nuinericul (;ul(;ulat,ions, while Laplace left this decidedl}^
important aspect out of consideration altogether. It remained in
reality to C'hai-iier to fiunish the Laplacean doctrine with a prac-
tical method for comi)uting the various statistical parameters.
And in the meantime the Gaussian methods reigned supreme
while Laiila('(^'s gi-eat woi'k was neglected.
The car(>ful rc^ader will readily notice that in the treatment of
frequency cnives I have allowed the semi-invariants, originally
introduc(xl in the theory of statistics by Thiele, to occupy a central
position. In my opinion the semi-invariants represent a more
powerful tool than the method of moments. I have also tried to
rescue from oblivion the important and original memoir by the
Danish actuary, Gram, and give to him and the French math-
ematician, Ilermite, their due recognition as the earliest investi-
gators of skew frequency distributions. Gram was perhaps the
first investigator to make proper use of the orthogonal functional
properties of the Lai)lacean normal frequency curve and its deriva-
tives. By means of an application of the orthogonal properties of
the Hermite polynomials and their close relation to the theory of
integral equations, tlu^ \\\\o\v theory of frequency distribution
can be presented in a d(>('idedly compact form; and I deem no
apology necessary for having introduced in my treatment of
frequency curves some of tlu^ more elementary theorems of integral
equations, that youngest branch of higher analysis, which at
present occupies a central position in advanced mathematics.
The most recent investigations along those lin(\s have been
made by the Swedish astronomer, Charlier, and his disciples,
Jorgensen and Wicksell. Unfortunately these investigations have
hitherto not received adequate and systematic treatment in Eng-
lish and American texts on statistics, and it is my hope that
the following pages may be of service in opening the eyes of
English speaking statisticians to the practical utility of these
methods.
The examples have all been selected so as to give a complete and
detailed illustration of the application of the theory to essentially
practical problems. I hav(>, on the other hand, purposely refrained
from giving the customary exercises, so-called, usually found in
statistical texts, especially those in German and English.
Although I have been a close student of and have read most of
the published statistical text-books in about seven languages for
the last ten j^ears, I regret to state that I have found little or no
XU PREFACE TO THE SECOND EDITION.
practical \a\uv in such trick exei'cises, which as a rule have but
slight relation to problems occurring in daily life.
Since lh(^ appearance of the first edition of this book in 1916
a nunil)er of exc(>llent statistical texts have been issued. Among
these I may mention a new edition of Yule's well-known ele-
mentary text, a greatl}' enlarged (xlition of Bowle3''s FAements of
Statistics, the new treatise by Caradog Jones, an enlarged German
translation of Charlier's Grunddragen, a very lucid Swedish text by
Wicksell, the scholarly and broadly planned Statistikens Teori i
Grundrids (in Danish) by Westergaard, and last but not least, the
thesis b}' Jorgensen, Frekvensflader og Korrelaiion}
Although an extended residence in the United States has per-
haps imi^roved my barbaric Dano-English, I fear that I must still
apologize to the reader for my shortcomings in rhetoric and gram-
mar. Most of the serious defects have, I hope, been overcome by
the diligent efforts of my co-editor and translator. Miss C. Dickson,
mathematical assistant in the department of Development and
Research of the American Telephone and Telegraph Company,
jNIiss Dickson's work has indeed been much beyond that of mere
translation. Her knowledge of the mathematical theory of prob-
abilities has enabled her to suggest to me several improvements in
my Danish notes.
I am also under great obHgations to a number of friends and
colleagues who have assisted me in the preparation of this volume.
I am especially indebted to Mr. E. C Molina, the well-known
jirobability expert of the American Telephone and Telegraph
Company. Mr. Molina's extensive knowledge of the works
of the old French masters, especially of those of Laplace,
has been of the greatest value to me, and I can truthfully
say that I have nowhere met a mathematician so thoroughly
acquainted with the intricacies of the Theorie Anahjtiquc as
Mr. Molina.
My thanks are also due to Mr. F. L. Hoffman, the Statistician
of the Prudential Insurance Compan}^, for the interest he took in
ni>- work along those lines while I was employed as a computer in
his department. To Messrs. M. C. Rorty and D. R. Belcher of the
Ameiican Telephone and Telegraph Company, I beg leave to
' As a pure probability text we may mention G. Castelnuovo's, Calculo delle
Prohabilita (Milano, 1919), as an exceptionally lucid and rifz;oi-ou3 treatise.
The recently issued Treatise on Prohnhilily by J. M. Keynes is briefly discussed
in paragraph 138 of this book. A. F.
PREFA{ E TO THE SECOND EDITION. XIU
express my best thanks for th(ur kind advice and encouragement
in the preparation of this volume.
It is indeed impossible to adequately express in a mere formal
preface my ol)ligations to Mr. Ilorty in this matter. His introduc-
tory note I regard as one of the highest rewards I have received in
this field of endeavor where one must usually be content with the
appreciation of one's peers. In this connection it is of interest to
note that Mr. Rorty is the pioneer investigator in the application
of the mathematical theory of probabilities to telephone engineer-
ing, which has been further developed in recent years by Molina of
America, Erlang and Johannsen of Denmark, Holm of Sweden,
Odell and Clrinsted of Great Britain. The pioneer work by Mr.
Rorty in this eminently practical fi(>ld antedates the earliest work
by L^rlang in Tidsskrift for Matemalik by nearly five years.
Last, but not least, I wish to convey my sincerest thanks to my
Scandinavian compatriots, Westergaard, Charlier, Jorgensen,
Wicksell and Guldberg from whose works I have drawn so freely.
To these gentlemen and to the works of the late Messrs. Gram and
Thiele of Copenhagen I really owe anything of value which may
be contained in this work.
Arne Fisher.
New York,
April, 1922.
INTRODUCTORY NOTE TO THE FIRST EDITIOX.
I feel it a great honor to have been asked by my friend and
colleague, Mr. Arne Fisher, of the Equitable Life Assurance
Society of the United States, to write an introductory note to
what appears to me the finest book as yet compiled in the English
language on the subject of which it treats. As an Examiner
myself in Statistical ^Method for a British Colonial Government,
it has been to me a heart-breaking experience, when implored by
intending candidates for examination to recommend a text-book
dealing with ]\Ir. Fisher's subject matter, that it has heretofore
been impossible for me to recommend one in the English language
which covers the whole of the ground. Until comparatively
recent years the case was even worse. While in P^rench, in Italian,
in German, in Danish, and in Dutch, scientific works on statistics
were available galore, the dearth of such hterature in the English
language was little short of a national or racial scandal. With
such works as those of Yule and Bowley, in recent years, there
has been some possibility for the English-speaking student to
acquire part of the knowledge needed. But it is harflly necessary
to point out what a very large amount of new ground is covered
by Mr. Fisher's new book as compared with such works as I have
referred to.
Despite my professional connection with statistical and actu-
arial work of a technical character my own personal interest in
]\Ir. Fisher's book is concentrated principalh' on the metaphysical
basis of the Probability-theory, and it is with regard to this
aspect of the subject alone that I feel qualified to comment on his
achievement. With all the controversy that has gone on through
many decades among metaphysicians and among writers on logic
interested especially in the bases of the theories of probability and
induction, between the pure empiricists of the type of J. S, ]\IilI
and John Venn (at all events in the earliest edition of his work)
on the one hand, and the (partly) a priori theorists who base their
doctrine on the foundation of Laplace on the other hand, it has
XVi IXTRODUCTORY XOTE TO THE FIRST EDITION.
been a source of intense satisfaction to me, as in the main a dis-
ciple of the latter group of theorists, to note the masterly way in
which Mr. Arne Fisher disentangles the issues which arise in the
keen and sometimes almost embittered controversy between these
two schools of thought. It has always seemed to the present
writer as if the very foundations of Epistcmology were involved
in this controversy. The impossibility of deriving the corpus of
human knowledge exclusively from emj)irical data by any logic-
ally valid process — an impossibility which led Immanuel Kant
to the creation of his epoch-making ])hilosophical system — is
hardly anywhere made more evident than in what seems to the
present writer the unsuccessful effort of thinkers like John Venn
to derive from such purely empirical data the entire Theory of
Probability, The logical fallacy of the process is analogous to
that perpetrated by John Stuart ^Nlill in endeavoring to base the
Law of Causality on what he termed an " induct io per simpliceni
enumerationemy Probably there is nowhere a more trenchant
and conclusive exposure of the unsoundness of this point of view,
than in the Right Honorable Arthur James Balfour's monu-
mental work "A Defense of Philosophic Doubt." It is there-
fore satisfactory to find that IVIr. Fisher emphasizes, quite at the
beginning of his treatise, that an a priori foundation for " Proba-
bility " judgments is indispensable.
Hardly less gratifying, from the metaphysical point of view,
is Mr, Fisher's treatment of the celebrated quaestio vexata of
Inverse Probabilities and his qualified vindication of Bayes'
Rule against its modern detractors.
Aside altogether from metaphysics, it is particularly satis-
factory to note the full and clear way in which the author treats
the Lexian Theory of Dispersion and of the "Stability" of sta-
tistical series and the extension of this theory by recent Scandi-
navian and Russian investigators, — a branch of the science which
has till the ajjpca ranee of this new work not been adequately
covered in English text-books.
It may of course be a moot question whether the preference
given by our author to Charlier's method of treating " Frequency
Curves" over the method of Professor Karl Pearson is well
advised. But whatever the experts' verdict may be on debatable
INTRODUCTORY NOTE TO THE FIRST EDITION. XVll
questions like these, the scientific world is to be congratulated on
INIr. Fisher's presentment of a new and sound point of view, and
he emphatically is to be congratulated on the production of a
text-book which for many years to come will be invaluable both
to students and to his confreres who are engaged in extending
the boundaries of this fascinating science.
F. W. Frankland,
Member of the Actuarial Society of America,
Felloiv of the Institute of Actuaries of Great
Britain and Ireland, and Felloiv of the
Royal Statistical Society of London.
New York,
October 1, 1915.
PREFACr: TO THE FIRST EDITION.
" Probal)ility " lias lono- ago ceased to be a mere theory of games
of cliance and is everywhere, esi)ecially on tlie continent, regarded
as one of the most important branches of api)Hed mathematics.
This is proven by the increasing number of standard text-books in
French, German, Itahan, Scan(Hnavian and Russian which have
appeared during the hist ten years. 1 )uring this time the research
work in the tlieory of probabihties lias receive<l a new impetus
through the hibors of the Enghsh biometricians under the leader-
ship of Pearson, the Scandinavian statisticians Westergaard,
Charlier and Kiji^r, the German statistical school under Lexis, and
the brilliant investigations of the Russian school of statisticians.
Each group of these investigations seems, however, to have
moved along its own particular lines. The English schools have
mostly limited their investigations to the field of biology as pub-
lished in the extensi^'e memoirs in the highly specialized journal,
Biometrika. The Scandinavian scholars have produced researches
of a more general character, but most of these researches are un-
fortunately contained in Scandinavian scientific journals and are
for this reason out of reach to the great majority of readers who
are not familiar with any of the allied Scandinavian languages.
This applies in a still greater degree to the Russians. German
scholars of the Lexis school have also contributed important
memoirs, but strangely enough their researches are little known
in this country or in England, a fact which is emphasized through
the belated English discussion on the theory of dispersion as devel-
oped by Lexis and his (lisci])les. The same can also be said with
regard to the Italian statisticians.
In the present work I have attempted to treat all these modern
researches from a common point of view, based upon the mathe-
matical principles as contained in the immortal work of the great
Laplace, "Theorie analyticjue des Probabilites," a work which
despite its age remains the most important contribution to the
xL\
XX PKEFACK TO THE FIRST EDITION.
theory of probabilities to our present day. Charlier has rightly
observed that the modern statistical methods may be based upon
a few condensed rules contained in the great work of Laplace.
This holds true des])ite the fact that many modern English
writers of late have shown a certain distrust, not to say actual
hostility, towards the so-called mathematical probabilities as
defined by the French savant, and have in their place adopted the
purely empirical probability ratios as defined by IMill, Venn and
Chrystal. It is quite true that it is possible to build a consistent
theory of such ratios, as for an instance is done by the Danish
astronomer and actuary, Thiele. The theory, however, then
becomes purely a theory of observations in which the theory of
probability takes a secondary place. The distrust in the so-called
mathematical a priori probabilities of Laplace I believe, however,
to be unfounded, and the criticism to which that particular kind
of probabilities is subjected by a few of the modern English
writers is, I believe, due to a misapprehension of the true nature
of the Bernoullian Theorem. This renowned theorem remains
to-day the cornerstone of the theory of statistics, and upon it I
have based the most important chapters of the present work.
Following the beautiful investigations of TschebychefT and
Pizetti in their proofs of Bernoulli's Theorem and the closely
related theorem of large numbers by Poisson I have adopted the
methods of the Swedish astronomer and statistician, Charlier,
in the discussion of the Lexian dispersion theory.
The theory of frequency curves is treated from various points
of view. I have first given a short historical introduction to the
various investigations of the law of errors. The Gaussian
normal curve of error was by the older school of statisticians
held to be sufficient to represent all statistical frequencies, and
actual observed deviations from the normal curve were attributed
to the limited lunnber of observations. Through the original
memoirs of Lexis and the investigations of Thiele the fallacy of
such a dogmatic belief was finally sliown. The researches of
Thiele, and later of Pearson, developed Inter tli(> theory of skew
curves of error. As recently as 1905 Charlier finally showed
that the whole theory of errors or frequency curves may be
brought back to the principles of Laplace. I have treated this
PREFACr: TO THE FIRST EDITION. XXI
subject by the methods of both PearsoM and Charlier, although I
have given tlie methods of t!ie hitter a ])re<h)minant jilace, because
of their easy and sim])le ap])Hcation in the])raetieal computations
required by statistical work. The mathematical theory of cor-
relation, which is tr(^ated in an elementary manner only, is based
upon the same ])rinci])les.
The statistical cxanii)lcs serve as illustrations of the theory, and
it will be noted that it is ])ossible to solve all the im])()rtant sta-
tistical })r()bh'ms ])resentiiig themselves in daily work on the basis
of a theory of mathematical probabilities instead of on a direct
theory of statistical metliods. I have here again followed Charlier
in dividing all statistical problems into two distinct groups,
namely, the homograde and th(> heterograde groups.
In treating the ])hil()S()i)hical side of the subject I have naturally
not gone into much detail. However, I have tried to emphasize
the two diametrically o})posite standpoints, namely the principle
of what von Kries has called the principle of "cogent reason,"
and the principle which Boole has aptly termed "the equal
distribution of ignorance." These two principles are clearly illus-
trated in the case of the so-called inverse probabilities. As far as
pure theory is concerned, the theory of "inverse probabilities"
is rigorous enough. It is only when making practical applications
of the rule of inverse probabilities (the so-called Bayes' Rule)
that many writers have made a fatal mistake by tacitly assuming
the principle of " insufficient reason " as the only true rule of com-
putation. Thisleads to paradoxical results as illustrated by the
practical problem from the region of actuarial science in Chapter
VI in this book.
In a work of this character I ha\e naturally made an extended
use of the higher mathematical analysis. However, the reader
wdio is not versed in these higher methods need not feel alarmed
on this account, as the elementary chapters are arranged in such a
way that the more difficult paragraphs may be left out. I have
in fact divided the treatise into two separate parts. The first
part embraces the mathematical probabilities proper and their
applications to homograde statistical series. This i)art, I think,
constitutes what is usually given as a course in vital statistics in
many American colleges, I hardly deem it worth while to give a
XXll PREFACE TO THE FIRST EDITIOX.
detailed discussion on the collection and arrangement of the sta-
tistical data as to various frequency distributions. The mere
graphical and serial representation of frequency functions by
means of histographs and frequency columns is so sim])le and
evident that a detailed description seems superfluous. The fitting
of the various curves to analytical formulas and the determination
of the various parameters seem to me of much greater impor-
tance. The theory of curve fitting which is treated in the second
volume is founded upon a more advanced mathematical analysis
and is for this reason out of reach to the average /American student
who desires to learn only the rudiments of modern statistical
methods. Practical statisticians, on the other hand, will derive
much benefit from these higher methods. It is a fact generally
noted in mathematics that the practical application of a difficult
theory is much simpler than that of a more elementary theory.
This is amply proven by the appearance of an excellent little
Scandinavian brochure by Charlier: "Grunddragen af den mate-
matiske Statistikken." ("Rudiments of Mathematical Statis-
tics.") I have always attempted to adapt theory to actual
practical problems and requirements rather than to give a purely
mathematical abstract discussion. In fact it has been my aim
to present a theory of probabilities as developed in recent years
which would ])rove of value to the practical statistician, the
actuary, the biologist, the engineer and the medical man, as
well as to the student who studies mathematics for the sake of
mathematics alone.
The nucleus of this work consisted of a number of notes written
in Danish on various aspects of the theory of })robabilities, col-
lected from a great number of mathematical, philosophical and
economic writings in various languages. At the suggestion of
my former esteemed chief, Mr. H. W. Robertson, P\A.S., As-
sistant Actuary of the Equitable Life Assurance Society of the
United States, I was encouraged to collect these fragmentary
notes in systematic form. The rendering In l-'nglish was done
by myself personally with the assistance of ^Ir. W. Bonynge.
With his assistance most of the idiomatic errors due to my
barbaric Dano-English have been eliminated. The notes stand,
however. In tli(> main as a faithful reproduction of my original
PREFACK TO THE FIRST EDITIOX. XXUl
English copy. Although the resulting " Duno-English " may-
have its great shortcomings as to rlietoric aiul grammar, I hope
to have succeeded in exj)ressing what I wanted to say in such
a manner that my possible readers may follow me without
difficulty.
I gladly take the opi)()rtunity of expressing my tjianks to a
number of friends and colleagues who in various ways have as-
sisted me in the preparation of this work. ]My most grateful
thanks are due to ]Mr. F. W. Frankland, Mr. II. \V. llobertson
and Mr. Wm. Bonynge not only for reading the manuscript and
most of the proofs, but also for the friendly help and encourage-
ment in the completion of this volume. The introductory note
by Mr. Frankland, coming from the pen of a scholar who for the
most of a life-time has worked with statistical-mathematical
subjects and who has taken a special interest in the i)hilosophical
and metaphysical aspects of the probability theory, I regard as
one of the strong points of the book. My debts to Messrs.
Frankland and Robertson as well as to Dr. W. Strong, Associate
Actuary of the Mutual Life Insurance Company, are indeed of
such a nature that they cannot be expressed in a formal preface.
My thanks are also due to Mr. A. Pettigrew in correcting the
first rough draught of the first three chapters at a time when my
knowledge of English was most rudimentary, to ]\Ir. ]\I. Dawson,
Consulting Actuary, and ^Ir. R. Henderson, Actuary of the Equit-
able Life, for reading a few chapters in manuscript and making
certain critical suggestions, to Professors C. Grove and W. Fite, of
Columbia University, for lumierous technical hints in the working
out of various mathematical formulas in Chapter VI, to Miss
G. Morse, librarian of the Equitable Library, in the search of
certain bibliographical material. Last but not least I wish to
express my sincerest thanks to several of my Scandinavian com-
patriots for allowing me to quote and use their researches on
various statistical subjects. I want in this connection especially
to mention Professor Charlier, of Lund, and Professors Wester-
gaard and Johannsen, of Copenhagen.
To The ^lacmillan Company and The New Era Printing Com-
pany I beg leave to convey my sincere appreciation of their very
courteous and accommodating attitude in the manufacture of
XXIV PREFACE TO THE FIRST EDITION.
this work. Their spirit has been far from commercial in this —
from a pure business standpoint — somewhat doubtful under-
taking.
Arne Fisher.
New York,
October, 1915.
TABLE OF CONTENTS.
PART I.
MATHEMATICAL PROBABILITIES AND IIOMOGRADE
STATISTICS.
Chapter I.
Introduction: General Principles and Philosophical Aspects.
1. Methods of Attack ^""^^
2. Law of C'ausality ,
3. Hyi)othoticaI Ju(lj:;niont.s o
4. Hj'pothofical Disjunctive .ludgnients 4
5. General Definition of the Probability of an Event .5
6. Equally likely Cases g
7. Objective and Subjective Probabilities o
Chapter II.
Historical and Bibliographical Notes.
S. Pioneer Writers ■,-.
9. Bernoulli, de Moivre and Bayes j9
10. Application to Statistical Data jo
11. Laplace and Modern Writers -.a
Chapter III.
The Mathematical Theory of Probabilities.
12. Definition of Mathematical Probability jy
13. Example 1 lo
14. Example 2 20
15. Example 3 20
16. Example 5 22
17. Example G 23
Chapter IV.
The Addition arid Multiplication Theorems in Probabilities.
18. Systematic Treatment by Laplace 26
19. Definition of Technical Terms ^a
20. The Theorem of the Compl(>te or Total Probability, or the Proba-
bility of "Either Or" r,j
21. Theorem of the Comi)ound Probability or the Probability of "As
Well As" 28
22. Poincarc's Proof of the Addition and Multiplication Theorem 30
23. Relative Probabilities gj
24. Multiplication Theorem 32
25. Probabilit}' of Repetitions 23
1* XXV
XXVI TABLE OF CONTENTS.
26. Application of the Addition and Multiplication Theorems in Problems
in Probabilities 35
27. Example 12 35
28. Example 13 36
29. Example U 37
30. Example 1.5 37
31. Example 16 38
32. E.xample 17 39
33. Example IS. De Moivre's Problem 40
34. Example 19 42
35. Example 20. Tchebycheff 's Problem 46
Chapter V.
Mathematical Expectation.
36. Definition, ]Mean Values 49
37. The Petrograd (St. Petersburg) Problem 51
38. Various Explanations of the Paradox. The Moral Expectation. ... 51
Chapter VI.
Probability a Posteriori.
39. Bayes's Rule. A Posteriori Pr<)l)abilities 54
40. Discovery and History of the Rule 55
41. Bayes's Rule (Case I) 56
42. Bayes's Rule (Case II) 59
43. Determination of the Probabilitic^s of Future Events Based upon
Actual Observations 59
44. Examples on the Application of Bayes's Rule 61
45. Criticism of Bayes's Rule 02
46. Theory versus Practice 64
47. Probabilities expressed by Integrals 67
48. Example 24 70
49. Example 25. Bing's Paradox 72
50. Conclusion 76
Chapter VII.
The Law of Large Numbers.
51. A Priori and Empirical Probabilities 82
52. Extent and Usage of Both Methods 85
53. Average a Priori Probabilities 87
54. The Theory of Disjjcrsion 88
55. Historical Development of the Law of Large Numbers 89
Chapter VIII.
Introductory Formulas from the Infinitesimal Calculus.
56. Special Integrals 90
57. Wallis's Expression of tt as an Infinite Product 90
58. De Moivre — Stirling's Formula 92
TABLE OF CONTENTS. XXvii
Chapter IX.
Law of Large Numbers. Mathematical Deduction.
59. Repeated Trials 96
60. Most Probable Value 97
61. Simple Numerical Examples 97
62. The Most Probable Value in a Series of Repeated Trials 99
63. Approximate Calculation of the Maximum Term, T,,, 101
64. Expected or Probable Value 102
65. Summation Method of Lajjlace. The Mean Error 104
66. Mean Error of Various Algebraic Expressions 106
67. Tchebycheff's Theorem 108
68. The Theorems of Poisson and Bernoulli proved by the Application
of the Tchebycheffian Criterion 110
69. Bernoullian Scheme 110
70. Poisson's Scheme Ill
71. Relation between Empirical Frequency Ratios and Mathematical
Probabilities 114
72. Application of the Tchebycheffian Criterion 115
Chapter X.
The Theory of Dispersion and the Criterions of Lexis and Charlier.
73. Bernoullian, Poisson and Lexis Series 117
74. The Mean and Dispersion 118
74a. Mean or Average Deviation 122
75. The Lexian Ratio and Charlier Coefficient of Disturbancy 124
Chapter XI.
Application to Games of Chance and Statistical Problems.
76. Correlate between Theory and Practice 127
77. Homograde and Heterograde Series. Technical Terms 128
78. Computation of the Mean and the Dispersion in Practice 130
79. Westergaard's Experiments 136
80. Charlier's Experiments I37
81. Experiments by Bonynge and Fisher 141
CHAPTER XII.
Continuation of the Application of the Theory of ProbdbUities to
Homograde Statistical Series.
82. General Remarks 146
83. Analogy between Statistical Data and Mathematical ProbabiUties . . 147
84. Number of Comparison and Proportional Factors 149
85. Child Births in Sweden 151
86. Child Births in Denmark 152
XXV (11 TABLE OF CONTENTS.
87. Danish Marriage Series 153
88. Stillbirths 154
89. Coal Mine Fatalities 155
90. Hcdiieod and Weighted Series in Statistics 157
91. Secular and Periodical Fluctuations 161
92. Cancer Statistics 165
93. Application of the Lexian Dispersion Theory in Actuarial Science.
Conclusion 167
PART II.
FREQUENCY CURVES AND HETEROGRADE
STATISTICS
Ch.vpter XIII.
The Theory of Errors and Frequency Curves and Its Application
to Statistical Series. General Remarks.
94. General Remarks. The Hypotheses of Elementary Errors 169
95. Application to Statistical Series. Definitions 173
96. (yoni[)ound Frequency Curves 176
97. Early Writers 178
98. Laplace and Gauss 179
99. (^uetelet's Studies 181
100. Opperman, Gram, and Thiele 182
101. Modern Investigations 184
Chapter XIV.
The Mathematical Theory of Frequency Curves.
102. Frc(iuency Distributions 188
103. Parameters Considered as Symmetric Functions 189
104. Semi-Invariants of Thiele 191
105. '^l'h(> I'\)uri(>r Integral Ecjuation 194
106. Frequency Function as the Solut ion of an Integral Equation 195
107. The Normal or Laplaccan Proba!)ility Function 197
108. Hermite's Polynomials 199
109. Orthogonal Functions 200
110. The Frequency Function Expressed as a Series 202
111. Derivation of Gram's Series 203
112. Absolute Frequencies 206
113. Cocflicients Expres,scd by Semi-Invariants 208
11 1. ( liaiigc of Origin and Unit 210
TABLE OF CONTENTS. Xxix
PART in.
PRACTICAL APPLICATIONS OF THE THEORY.
Ciiaptp:k XV.
The Numerical Determination of the Parameters.
115. CU'iU'ial Remarks 215
1 IG. Remarks on Cri( ieisms 216
117. Charlier's Computation Scheme 218
118. Comparison between Observed Data and Theoretical Values 220
119. Principle of Method of Least Squares 221
120. Gau.ss' Solution of Normal Equations 224
121. Arithmetical Ai)pIication of Method 225
Chapter XVI.
Logarithmically Transformed Frequency Functions.
122. Transformalion of the Variate 235
123. The General Theory of Transformation 236
124. Lo^aritlimic; Transformation 237
125. The Mathematical Zero 238
125. Logarithmically Transformed Frequency Series 239
127. Parameters DeterminiMl by Least Squares 243
128. Application to Graduation of Mortality Tables 244
129. Formation of Observation Equations 246
130. Additional Examples ; 257
ClIAlTKlt Xv'H.
Frequency Curves and their Relation to the Bernoullian Series.
131. The Bernoullian Series 261
132. Poisson's Exponent ial 265
133. The Law of Small Numbers 270
Chapter XVIII.
Poisson-Charlier Frequency Curves for Integral Variates.
134. Charlier's B Curve 271
135. Numerical Examples 273
136. Transformation of the Variate 274
137. Bernoullian Series (>xpressed as B Curves 27t)
138. Remarks on Mr. Kevnes' Criticisms 278
PART I
MATHEMATICAL PROBABILITIES AND
HOMOGllADE STATISTICS
C'lIAITKR I.
INTRODUCTION: GENERAl, 1MU.\( Il'l.KS AND PHILOSOPHICAL
ASPIX'TS.
1. Methods of Attack. The subjrct of the tlieory of proba-
bilities may be attacked iii two dilfereiit ways, namely in a
philosophical, and in a mathematical manner. At first the
subject originated as isolated mathematical ])roblems from games
of chance. The pioneer writers on ])robability such as Cardano,
Galileo, Pascal, Fermat, and Iluyghens treated it in this way.
The famous Bernoulli was, perhaps, the first to view the subject
from the philosopher's point of \iew. Laplace wrote his well-
known "Essai Philosophiciue des Probabilites," wherein he terms
the whole science of i)r()bability as the ai)j)lication of common
sense. During the last thirty years numerous eminent jjliilo-
sophical scholars such as Mill, Vemi, and Keynes of England,
Bertrand and Poincare of France, Sigwart, von Kries and Lange
of Germany, Kroman of Denmark, and several Russian scholars
have written on the ])hilosophical aspect.
In the ordinary presentation of the elements of the theory of
probability as found in most English text-books, the treatment
is wholly mathematical. The student is given the definition of
a mathematical probability and the elementary theorems are
then proved. ^Ye shall, in the following chapter, depart from
this rule and first view the stibject, briefly, from a philosophical
standpoint. What the student may thus lose in time we hope
he may gain in obtaining a broader view of the ftmdamental
principles underlying our science. At the same time, the reader
who is unacquainted with the science of philosophy or pure logic,
need not feel alarmed, since not even the most elementary
knowledge of the principles of formal logic is required for the
understanding of the following chapter.
2. Law of Causality. — In a great treatise on the Chinese civiliza-
tion, Oscar Peschel, the German geographer and philosopher,
makes the following remarks: "Since otir intellectual awakening,
since we have appeared on the arena of history as the creators
2 1
^ INTRODUCTION. [2
and guardians of the treasures of cidture, we have sought after
only one thhig, of the presence of which the Chinese had no
i(k'a, and for which they would give hardly a bowl of rice. This
invisible thing we call causality. \Ye have admired a vast
nund)cr of Chinese inventions, but even if we seek through their
huge treasures of philosophical writing we are not indebted to
them for a single theory or a single glance into the relation
between cause and effect."
The law of causality may be stated broadly as follows: Every-
thing that happens, and everything that exists, necessarily
happens or exists as the consequence of a previous state of things.
This law cannot be proven. It must be taken, a priori, as an
axiom; but once accepted as a truth it does away with the belief
of a capricious ruling power, and even if the strongest disbeliever
of the law may deny its truth in theory he invariably applies it
in practice during his daily occupation in life.
All future human activity is more or less influenced by past
and present conditions. Modern historical writings, as for
instance the works of the brilliant Italian historian, Ferrero,
always seek to connect past events with ])resent social and
economic conditions. Likewise great and constructive statesmen
in trying to shape the destinies of nations always reckon with
past and present events and conditions. We often hear the term,
"a man with foresight," applied to leading financiers and states-
men. This does not mean that such men are gifted with a vision
of the future, but simply that they, with a detailed and thorough
knowledge of past and present events, associated with the par-
ticular undertaking in which they are interested, have drawn
conclusions in regard to a future state of affairs. For example,
when the Canadian Pacific officials, in the early eighties, chose
Vancouver as the western terminal for the transcontinental
railroad, at a time when practically the whole site of the present
metropolis of western Canada was only a vast timber tract, they
realized that the conditions then i)r('\"ailing on lliis particular
spot — the excellent shipping facilities, the favorable location in
regard to the Oriental trade, and the natural wealth of the sur-
rounding country — would bring forth a great city, and their
predictions came true.
3] HYPOTHETICAL JUDGMENTS. 3
Predictions with regard to the future must be taken seriously
only when they are based upon a thorough knowledge of past
and present events and conditions. Prophecies, taken in a
purely biblical sense of the term and viewed from the law of
causality, are mere guesses which may come true and may not.
A prophet can hardly be called more than a successful guesser.
Whether there iuive been persons gifted with a purely prophetic
vision is a question which must be left to the theologians to
wrangle over.
3. Hypothetical Judgments. — Any person with ordinary in-
tellectual faculties may, however, predict certain future events
with absolute certainty by a simple application of the principle
of hypothetical judgment. The typical form of the hypothetical
judgment is as follows: If a certain condition exists, or if a certain
event takes place then another definite event will surely follow.
Or if A exists B will invariably follow.
iNIathematical theorems are examples of hypothetical judg-
ments. Thus in the geometry of the plane we start with certain
ideas (axioms) about the line and plane. From these axioms
we then deduce the theorems by mere hypothetical judgments.
Thus in the Euclidian geometry we find the axiom of parallel
lines, which assumes that through a point only one line can be
drawn parallel to another given line, and from this assumption
we then deduce the theorem that the sum of the angles in a
triangle is 180°. But it must be borne in mind that this proof is
valid only on the assumption of the actual existence of such lines.
If we could prove directly by logical reasoning or by actual
measurement, that the sum of the angles in any triangle is equal
to 180°, then we would be able to prove the above theorem, the
so-called "hole in geometry," independently of the axiom of
parallel lines.
A Russian mathematician, Lobatschewsky, on the other hand,
assumed that through a single point an infinite number of parallels
might be drawn to a previously given line, and from this as-
sumption he built up a complete and valid geometry of his own.
Still another mathematician, Riemann, assumed that no lines
were parallel to each other, and from this produced a perfectly
valid surface geometry of the sphere.
4 INTRODUCTION. [4
As exam])les of hypothetical judj;inent we have the twia follow-
ing well-known theorems from elementary geometry and algebra.
If one of the angles of a triangle is divided into two parts, then
the line of division intersects the opposite side. If a dccadian
number is dixided by ') there is no remainder from the division.
In natural science, hy])()thetical judgments are founded on
certain occurrences (phenomena) which, without exception, have
taken place in the same manner, as shown by repeated obser-
vations. The statement that a suspended body will fall when its
supi)()rt is removed is a hy])othctical judgment derived from
actual ex])crieiice and o})servati()n.
4. Hypothetical Disjunctive Judgments. — In hypothetical
judgments we arc always able to associate cause and effect. It
happens frequently, however, that our knowledge of a certain
comj^lcx of present conditions and actions is such that we are
not able to tell beforehand the resulting consequences or effects
of such conditions and actions, but are able to state only
that either an event A\ or an event E2, etc., or an event En will
happen. This represents a hypothetical disjunctive judgment
whose typical form is: If A exists either Ei, E-i, E3, • • • or En
will happen.
If we take a die, ?'. e., a homogeneous cube whose faces are
marked with the numbers from one to six, and make an ordinary
throw, we are not able to tell beforehand which side will turn
up. True, we have here again a previous state of things, but the
conditions do not allow such a simple analysis as the cases we
have hitherto considered under the ])urely hypothetical judgment.
Here a multitude of causes influence the final result — the weight
and centre of gravity of the die, the infinite number of possible
movements of the hand which throws the die, the force of contact
with which the die strikes the table, the friction, etc. All these
causes are so complex that our minds are not allordcd an Op_
])ortunity to gras]) and distinguish the im])ulses that determine
the fall of the die. In other words we are not able to say, a
])riori, which face will apj)ear. We onl>' kiu)W for certain that
either 1, 2, 'A, 4, ."), or (> will appear. If a line is drawn through
the vertex of a triangle, it either intersects the opposite side or
it does not. If a number is divided bv 5 the division either gives
5] GENERAL DEFINITION OF PROBABILITY OF AN EVENT. 5
only an integral number or leaves a remainder. If an opening
is made in the wall of a vessel partly filled with water, then either
the water escapes or remains in the vessel. All the above cases
are examples of hyi)otlK'tical disjunctive judgments.
The four cases show, however, a common characteristic. They
all have a certain partial domain, where one of the mutually
exclusive events is certain to happen, while the other partial
domain will bring forth the other event, and the total area of
action embraces both events. Taking the triangle, we notice
that the lines may pass through all the points inside of an angle
of 300°, but only the lines falling inside the internal vertical
angle, ip, of the triangle will produce the event in question,
namely the line intersecting the opposite side. There will be
an outflow from the vessel only if the hole is made in that part
of the wall which is touched by the fluid.
All problems do not allow of such simple analysis, however,
as will be seen from the following example. Suppose we have
an urn containing 1 white and 2 black balls and let a person
draw one from the urn. The hypothetical disjunctive judgment
immediately tells us that the ball will be either black or white,
but the particular domain of each event cannot be limited to the
fixed border lines of the former examples. Any one of the balls
may occupy an infinite numl)er of positions, and furthermore we
may imagine an infinite number of movements of the hand which
draws the ball, each movement being associated with a particular
point of position of the ball in the urn. If we now assume each
of the three balls to have occupied all possible positions in the
urn, each point of position being associated with its proper
movement of the hand, it is readily seen that a black ball will
be encountered twice as often as a white ball in a particular
point of position in the urn, and for this reason any particular
movement of the hand which leads to this point of position
grasps a black ball twice as often as a white ball.
5. General Definition of the Probability of an Event. — All the
above examples have shown the following characteristics:
(1) A total general region or area of action in which all actions
may take place, this total area being associated with all possible
events.
6 INTRODUCTION. [ 6
(2) A limited special domain in which the associated actions
produce a special event only.
If these areas and domains, as in the above cases, are of such a
nature that they allow a purely quantitative determination,
they may be treated by mathematical analysis. We define
now, without entering further into its particular logical signifi-
cance, the ratio of the second special and limited domain to the
first total region or area as the probability of the happening of
the event, E, associated with domain Xo. 2.
We must, however, hasten to remark that it is only in a com-
paratively few cases that we are able, a priori, to make such a
segregation of domains of actions. This may be possible in
purely abstract examples, as for instance in the example of the
division of the decadian number by 5. But in all cases where
organic life enters as a dominant factor we are unable to make such
sharp distinctions. If we were asked to determine the proba-
bility of an .r-y ear-old person being alive one year from now, we
should be able to form the hypothetical disjunctive judgment:
An .r-year-old person will be either alive or dead one year from
now. But a further segregation into special domains as was
the case with the balls in the urn is not possible. Many ex-
tremely complex causes enter into such a determination; the
health of the particular person, the surroundings, the daily life,
the climate, the social conditions, etc. Our only recourse in
such cases is to actual observation. By observing a large
number of persons of the same age, x, we may, in a ])urcly em-
pirical way, determine the rate of death or survival. Such a deter-
mination of an unknown probability is called an emi)irical proba-
bility. An empirical probability is thus a probability, into the
determination of which actual experience has entered as a domi-
nant factor.
6. Equally Likely Cases.^ — The main difficulty, in the appli-
cation of the ab()\-e definition of probability, lies in the deter-
mination of the question whether all the events or cases taking
place in the general area of action may be regarded as cciually
likely or not. Two diametrically ()])i)osite views have here been
brought forward by writers on probabilities. Oiu> view is based
upon the principle which in logic is known as tlie principle of
6] EQUALLY LIKELY CASES. 7
"insufficient reason," \vhili> the other view is bused ui)on the
principle of "cogent reason." The chissical writers on the theory
of probability, such as Jacob Bernoulli and Laplace, base the
theory on the principle of insufficient reason exclusively. Thus
Bernoulli declares the six possible cases by the throw of a die to
be equally likely, since "on account of the equal form of all the
faces and on account of the homogeneous structure and equally
arranged weight of the die, there is no reason to assume that any
face should turn up in preference to any other." In one place
Laplace says that the possible cases are "cases of which we are
equally ignorant," and in another place, "we have no reason to
believe any particular case should hai)pen in i)reference to any
other."
The ()p])osite view, based on the principle of cogent reason,
has been strongly endorsed in an admirable little treatise by the
German scholar, Johannes von Kries.^ Von Kries requires, first
of all, as the main essential in a logical theory of probability,
that "the arrangement of the equally likely cases must have a
cogent reason and not be subject to arbitrary conditions."
\n several illustrative exam])les, von Kries shows how the
principle of insufficient reason may lead to different and paradox-
ical results. The following example will illustrate the main
points in von Kries's criticism. Suppose we be given the follow-
ing problem: Determine the probability of the existence of human
beings on the planet ^Nlars. By applying the first mentioned
principle our reasoning would be as follows: We have no more
reason to assume the actual existence of man on the planet than
the complete absence. Hence the probability for the non-
existence of a human being, is equal to ^. Next we ask for the
probability of the presence or non-presence of another earthly
mammal, say the elephant. The answer is the same, h. Xow
the probability for the absence of both man and elephant on the
planet is ^ X § = 4."^ The ])robabilit\' for the absence of a third
mammal, the horse, is also 2, or the probability for the absence
of man, elephant, and horse is equal to (^)^ = |. Proceeding in
the same manner for all mammals we obtain a very small proba-
1 "Die Princij)ien dor '\^'allrs^heinlichkeitsrechnung."l Berlin, 1886.
2 See the chapter 011 multiplication of probabilities.
8 INTRODUCTION. [6
bility for the complete absence of all mammals on ]\Iars, or a
very large probability, almost equal to certainty, that the planet
harbors at least one mammal known on our planet, an answer
\vhich certainly does not seem plausible. But we might as well
have j)ut the question from the start: what is the probability
of the existence or absence of any one earthly mammal onlNIars?
The principle of insufficient reason when applied directly would
here give the answer §, while when applied in an indirect manner
the same method gave an answer very near to certainty.
An urn is known to contain white and black balls, but the
number of the balls of the two different colors is unknown. What
is the probability of drawing a white ball? The principle of
insufficient reason gives us readily the answer: ^, while the prin-
ciple of cogent reason would give the same answer only if it were
known a priori that there were equal numbers of balls of each
color in tlie urn before the drawing took place. Since this
knowledge is not present a priori, we are not able to give any
answer, and the problem is considered outside the domain of
probabilities. There is no doubt that the principle advocated
by von Kries is the only logical one to apply, and a recent
treatise on the theory of probability by Professor Bruhns of
Leipzig^ also gives the principle of cogent reason the most promi-
nent place. On the other hand it must be admitted that if the
principle was to be followed consistently in its very extreme it
would of course exclude many problems now found in treatises
on probability and limit the application of our theory consider-
ably in scope. Still, however, we must agree with von Kries
that it seems very foolhardy to assign cases of which we are
absolutely in the dark, as being equally likely to occur. This
very j)rinciple of insufficient reason is in very high degree re-
sponsible for the somewhat absurd answers to questions on the
so-called "inverse probabilities," a name which in itself is a great
misnomer. We shall later in the chapter on "a posteriori"
probabilities discuss this question in detail. At present we shall
only warn the student not to judge cases of which he has no
knowledge whatsoever to be equally likely to occur. The old rule
"experience is the best teacher" holds here, as everywhere else.
' " KoUektivmasslehre and WiihrHcheinlichkeitsrechnung," Loipzip, 1903.
7 J OBJECTIVE AND SUBJECTIVE PROBABILITIES, 9
7. Objective and Subjective Probabilities.— In tliis connection
it is interesting to note the lucid remarks by the Danish statis-
tician, Westergaard. "By every well arranged game of chance,
by lotteries, dice, etc.," Westergaard says, "everything is ar-
ranged in such a way that the causes influencing each draw or
throw remain constant as far as possible. The l)alls are of the
same size, of the same wood, and have the same density; they are
carefully mixed and each ball is thus apparently subject to the
influences of the same causes. However, this is not so. Despite
all our efforts the balls are different. It is impossible that they
are of exactly mathematically spherical form. Each ball has its
special deviation from the mathematical sphere, its special size
and weight. No ball is absolutely similar to any one of the
others. It is also impossible that they may be situated in the
same manner in tlie bag. In short there is a multitude of ap-
parently insignificant differences which determine that a certain
definite ball and none of the other balls may be drawn from the
bag. If such inequalities did not exist one of two things would
happen. Either all balls would turn up simultaneously or also
they would all remain in the bag. Many of these numerous
causes are so small that they perhaps are invisible to the naked
eye and completely escape all calculations, but by mutual
action they may nevertheless produce a visible result."
It thus appears that a rigorous application of the principle of
cogent reason seems impossible. However, a compromise
between this principle and that of the principle of insufficient
reason may be effected by the following definition of equally
possible cases, viz. : Equally possible cases are such cases in which
we, after an exhaustive analysis of the physical laws underlying the
structure of the complex of causes influencing the special event, are
led to assume that no particular case will occui in preference to any
other. True, this definition introduces a certain subjective
element and may therefore be criticized by those readers who
wish to make the whole theory of probabilities purely objective.
Yet it seems to me preferable to the strict application of the
principle of equal distribution of ignorance. Take again the
question of the probability of the existence of human beings on
the planet Mars. The principle of equal distribution of ignorance
10 INTRODUCTION. [7
readily gives us without further ado the answer ^. Modern astro-
physical researches luue, however, verified physical conditions on
the planet which make the presence of organic life quite possible,
and according to such an eminent authority as ]\lr. Lowell, perhaps
absolutely certain. Yet these physical investigations are as
yet not sufficiently complete, and not in such a form that they
may be subjected to a purely quantitative analysis as far as the
theory of probabilities is concerned. Viewed from the stand])oint
of the principle of cogent reason any attempt to determine the
numerical value of the above probability must therefore be put
aside as futile. This result, negative as it is, seems, however,
preferable to the absolute guess of | as the probability.
CHAPTER 11.
HISTORICAL AND BIBLIOCRAPIIICAL XOTES.
8. Pioneer Writers. — Tlie first attempt to dofino tlie measure
of a probahility of a future event is eredited to the Greek ])hilos-
oplier, Aristotle. Aristotle ealls an event probable when the
majority, or at least the majority of the most intellectual persons,
deem it likely to happen. This definition, although not allowing
a purely quantitative measurement, makes use of a subjective
judgment.
The first really mathematical treatment of chance, however, is
given by the two Italian mathematicians, Cardano and Galileo,
who both solved several problems relating to the game of dice.
Cardano, aside from his mathematical occupation, was also a
professional gambler and had evidently noticed that in all kinds
of gambling houses cheating was often resorted to. In order
that the gamester might be fortified against sucli cheating prac-
tices, Cardano wrote a little treatise on gambling wherein he
discussed several mathematical questions connected with the
diflFerent games of dice as ])Iayt>d in the Italian gambling houses
at that time. Galileo, although not a professional gambler, was
often consulted by a certain Italian nobleman on several problems
relating to the game of dice, and fortunately the great scholar
has left some of his investigations in a short memoir. In the
same manner the two great French mathematicians, Pascal and
Fermat, were often asked by a professional gamester, the cheva-
lier de Mere, to a])])ly their mathematical skill to the solution of
different gambling problems. It was this kind of investigation
which probably led Pascal to the discovery of the arithmetical
triangle, and the first rudiments of the combinatorial analysis,
which had its origin in ])r()bability problems, and which later
evolved into an independent branch of mathematical analysis.
One of the earliest works from the illustrious Dutch physicist,
Huyghens, is a small pamjjhlet entitled "de Ratiociniis in Ludo
Alese," printed in Leyden in the year 1657. Huyghens' tract is
11
12 HISTORICAL AND BIBLIOGRAPHICAL NOTES. [9
the first attempt of a systematic treatment of the subject. The
famous Leibnitz also wrote on chance. His first reference to a
mathematical probability is perhaps in a letter to the ])hiloso-
pher, Wolft", wherein he discusses the summation of the infinite
series 1 — 1 + 1 — IH----. Besides he solved several problems.
9. Bernoulli, de Moivre and Bayes. — The first extensive
treatise on the theory as a whole is from the hand of the famous
Jacob Bernoulli. Bernoulli's book, "Ars Conjectandi," marks a
revolution in the whole theory of chance. The author treats
the subject from the mathematical as well as from a philo-
sophical point of view, and shows the manifold applications of
the new science to practical problems. Among other important
theorems we here find the famous proposition which has become
known as the Bernoulli Theorem in the mathematical theory of
probabilities. Bernoulli's work has recently been translated
from the Latin into German,^ and a student who is interested in
the whole theory of probability should not fail to read this
masterly work.
The English mathematicians were the next to carry on the
investigations. Abraham de Moivre, a French Huguenot, and
one of the most remarkable mathematicians of his time, wrote
the first English treatise on probabilities.^ This book was cer-
tainly a worthy product of the masterful mind of its author, and
may, even today,, be read with useful results, although the
method of demonstration often appears lengthy to the student
who is accustomed to the powerful tools of modern analysis.
The high esteem in which the work by de ]Moivre is held by
modern writers, is proven by the fact that E. Czuber, the eminent
Austrian mathematician and actuary, so recently as two years
ago translated the book into German. A certain ])roblem (see
Chap. l\) still goes under the name of "The Problem of de
Moivre" in the modern literature on probability. A contem-
porary of de Moivre, Stirling, contributed also to the new branch
of mathematics, and his name also is immortalized in the theory
of probability by the formula which bears his name, and by which
we are able to express large factorials to a very accurate degree
of approximation. The third important English contributor is
' Ars Conjectandi, Ostwald's Klassiker No. 108, Leipzig, 1901.
^ de Moivre: "The Doctrine of Chances," London, 1781.
10] APPLICATION TO STATISTICAL DATA. 13
the Oxford clergyman, T. Bayes. Bayes' treatise, which was
published after his death by Price, in Philosophical Tratu'iactioiis
for 17G4, deals with the determination of the a posteriori proba-
bilities, and marks a very imj)()rtant stepping stone in our whole
theory, rnfortunately the rule known as Bayes' Rule has been
applied very carelessly, and that mostly by some of Bayes' own
countrymen; so the whole theory of Bayes has been repudi-
ated by certain modern writers. A recent contril)uti()n by the
Danish philosophical writer, Dr. Kroman, seems, however, to have
cleared up all doubts on the subject, and to have given Bayes his
proper credit.
10. Application to Statistical Data. — In the eighteenth century
some of the most celebrated mathematicians investigated
problems in the theory of probability. The birth of life as-
surance gave the whole theory an important application to
social problems and the increasing desire for the collection of all
kinds of statistical data by governmental bodies all over Europe
gave the mathematicians some highly interesting material to
which to apply their theories. No wonder, therefore, that we
in this period find the names of some of the most illustrious mathe-
maticians of that time, such as Daniel Bernoulli, Euler, Nicolas
and John Bernoulli, Simpson, D'Alembert and Buft'on, closely
connected with the solution of problems in the theory of mathe-
matical probabilities. We shall not attempt to gi\e an account
of the diherent works of these scientists, l)ut shall only dwell
briefly on the labors of Bernoulli and D'Alembert. In a memoir
in the St. Petersburg Academy, Daniel Bernoulli is the first to
discuss the so called St. Petersburg Problem, one of the most
hotly debated in the whole realm of our science. We may here
mention that this problem is today one of the main pillars in the
economic treatment of value Bernoulli introduced in the dis-
cussion of the above mentioned problem the idea of the "moral
expectation," which under slightly difl'erent names appears in
nearly all standard writings on economics.
D'Alembert is especially remembered for the critical attitude
he took towards the whole theory. Although one of the most
brilliant thinkers of his age, the versatile Frenchman made some
great blunders in his attempt to criticize the theories of chance.
14 HISTORICAL AND BIBLIOGRAPHICAL NOTES. [11
Biitfon's name is remein})ored because of the needle problem,
and he may properly be called the father of the so-called "ge-
ometrical" or "local" probabilities.
11. Laplace and Modern Writers. — We now come to that
resplendent genius in the investigation of the mathematical
theory of chance, the immortal Laplace, who in liis great work,
"Theorie Analytique des Probabilites," gave the final mathe-
matical treatment of the subject. This massive volume leaves
nothing to be desired and is still today — more than one hundred
years after its first publication — a most valuable mine of in-
formation and compares favorably with much more modern
treatises. But like all mines, it requires to be mined and is by
no means easy reading for a beginner. An elementary extract,
"Essai Philosophique des Probabilites," containing the more
elementary parts of Laplace's greater work and stripped of all
mathematical formulas has recently appeared in an English
translation.
Among later French works, Cournot's "Exposition de la
Theorie des Chances et des Probabilites" (1843), treated the
principal questions in the application of the theory to practical
problems in sociology. Li 1837 Poisson published his "Re-
cherches sur les Probabilites " in which he for the first time proved
the famous theorem which bears his name. Poisson and his
Belgian contemporary, Quetelet, made extensive use of the
theory in the treatment of statistical data.
Among the most recent French works, we mention especially
Bertrand's "Calcul des Probabihtes" (Paris, 1888), Poincare's
"Calcul des Probabilites" (Paris, 1896), and Borel's "Calcul des
Probabilites" (Paris, 1901). We especially recommend Poin-
care's brilliant little treatise to every student who masters the
French language, as this book makes no departure from the
lively and elucidating manner in which this able mathematical
writer treated the numerous subjects on which he wrote during
his long and brilliant career as a mathematician.
Of Russian writers, the mathematician, Tchebycheff, has given
some extensive general theorems relating to the law of large
numbers. Unfortunately Tchebycheff 's writings are for the
most part scattered in French, German, Scandinavian and
11] LAPLACE AND MODERN WRITERS. 15
Russian journals, and tlnis are not easily accessible to the ordinary
reader. A Russian artillery officer, Sabudski, has recently pub-
lished a treatise on ballistics in German, wherein he extends the
views formulated by Tchebycheff.
Of Scandinavian writers we mention T. X. Thiele, who prob-
ably was the first to ])ul)lish a systematic treatise on skew curves.^
An abridged edition of this very original work has recently been
translated into English.- The Dane, Westergaard, is the author
of the most extensive and thorough treatise on vital statistics
which we possess at the present time. Westergaard 's work has
recently been translated into German,^ and is strongly recom-
mended to the student of vital statistics on account of his clear
and attractive style of presenting this important subject.
The Swedish mathematicians Charlier and Gylden have
published a series of memoirs in different Scandinavian journals
and scientific transactions. We may also, in this category,
mention the numerous small articles by the eminent Danish
actuary. Dr. Gram.
While tlie German mathematicians in general are the most
fertile writers on almost every branch of pure and applied mathe-
matics, they have not shown much activity in the theory of
mathematical probability except in the past ten years. But
during that time there has appeared at least a dozen standard
works in German. Among these, the lucid and terse treatise
by E. Czuber, the Austrian actuary and mathematician, is
especially attractive to the beginner on account of the systematic
treatment of the whole subject.'* A very original treatment is
offered by H. Bruhns in his " Kollektivmasslehre und Wahrschein-
lichkeitsrechnung" (Leipzig, 1903). Among the German works,
we may also mention the book by Dr. Norman Herz in "Samm-
lung Schubert," and an excellent little work by Hack in the small
pocket edition of "Sammlung Goschen." The theory of skew
curves and correlation is presented by Lipps and Bruhns in
extensive treatises.
' "Almindelig lagttagclseslaere," Copenhagen, 1884.
2 "Theory of Observations," London, 1903.
3 "Mortalitat und MorbiHtat/' Jena, 1902.
*E. Czuber, "Wahrscheinlichkeitsrechnung," Leipzig, 1908 an<i 1910, 2
volumes.
16 HISTORICAL AND BIBLIOGRAPHICAL NOTES. [11
We finally come to modern iMiglish writers on the subject.
After the appearance of de ]\Ioivre's "Doctrine of Chances"
the first work of im])ortance was the book by de Morgan "An
Essay on the Theory of Probabilities." The latest text-book is
Whitworth's "Choice and Chance" (Oxford Press, 1904): but
none of these works, although A'ery excellent in their manner of
treatment of the subject, comes up to the French, Scandinavian,
and German text-books. Nevertheless, some of the most im-
portant contributions to the whole theory have been made by
the English statisticians and mathematicians, Crofton, Pearson,
and Edgeworth. Especially have frequency curves and cor-
relation methods introduced by Professor Karl Pearson been
very extensively used in direct applications to statistical and
biological problems. Of purely statistical writers, we may
mention G. Udny Yule, who has published a short treatise en-
titled "Theory of Statistics" (London, 1911). Numerous ex-
cellent memoirs have also appeared in the different English and
American mathematical journals and statistical periodicals,
especially in the quarterly publication, Biomctrika, edited by
Professor Karl Pearson.
In the above brief sketch, we have only mentioned the most
important contributors to the theory of i)robabilities proper.
Numerous able writers have written on the related subject of
least squares, the mathematical theory of statistics and insurance
mathematics. We shall not discuss the works of these inves-
tigators at the present stage. Each of the most important w'orks
in the above mentioned branches will receive a short review in
the corresponding chapters on statistics and assurance mathe-
matics. The readers interested in the historical develoi)ment of
the theory of probabilities are advised to consult the special
treatises on this subject by Todhunter and Czuber.^
1 After this chapter had gone to press I notice that a treatise by the emi-
nent English scholar, Mr. Keynes, is being prepared by The Macniillan Co.
In this connection I wish also to call attention to the recent publication by
Bachclior (Calcul des probabilites, 1912), a work planned on a broad and
extensive scale. — A. F.
CIIAlTKIl III.
THE MATHEMATICAL THEORY OF PROBABnJTIES.
12. Definition of Mathematical Probability. — '" If our positive
knowledge of the efl'ect of a complex of causes is such that we
may assume, a priori, t cases as being equally likely to occur, but
of which only/, (/ < t), cases are favorable in causing the event,
E, in which we are interested, then we define the proper fraction :
f,t = y as the mathematical pr()l)ability of the happening of
the event, £" (Czuber). We might also have defined an a
priori probability as the ratio of the equally favorable cases to
the co-ordinated possible cases.
As is readily seen, this definition assumes a certain a priori
knowledge of the possible and favorable conditions of the event
in question, and the probability thus defined is therefore called
"a priori probability." Denoting the event by the symbol, E,
we express tlie i)robability of its occurrence by the symbol F{E),
and the probability' of its non-occurrence by P{E). Thus if / is
the total number of equally possible cases and / the number of
favorable cases for the event, we have:
and
P{E) = j=V,
P(E) = ^-jJ= i-l=\-p=l- P(E).
This relation evidently gives us: P{E) + P(E) = 1, which is the
symbolic expression for the hypothetical disjunctive judgment
that the event E will either happen or not happen. If/ = /, we
have:
P(E) =1=1.
which is the symbol for the hypothetical judgment that if A
exists, E will surely happen. Similarly if/ = 0, we get
o 17
18 THE MATHEMATICAL THEORY OF PROBABILITIES. [ 13
P(E) = 7 = 0,
or the symbol for the hypothetical judgment: If A exists, E will
not happen, or what is the same, E will happen.
As we have already mentioned, in an a priori determination of
a probability, special stress must be laid upon the requirement
that all possible cases must be equally likely to occur. The
enumeration of these cases is by no means so easy as may appear
at first sight. Even in the most simple problems where there
can be doubt about the possible cases being equally likely to
occur, it is very easy to make a mistake, and some of the most
eminent mathematicians and most acute thinkers have drawn
erroneous conclusions in this respect. We shall give a few ex-
amples of such errors from the literature on the subject of the
theory of probabilities, not on account of their historical interest
alone, but also for the benefit of the novice who naturally is ex-
posed to such errors.
13. Example 1. — An Italian nobleman, a professional gambler
and an amateur mathematician, had, by continued observation
of a game with three dice, noticed that the sum of 10 appeared
more often than the sum of 9. He expressed his surprise at this
to Galileo and asked for an explanation. The nobleman re-
garded the following combinations as favorable for the throw of 9 :
1
2 6
1
3 5
1
4 4
2
2 5
2
3 4
3
3 3
id for the throw of 10 the six combinations of
1
3 6
1
4 5
2
2 6
2
3 5
2
4 4
3
3 4
13] EXAMPLE 1. 19
Galileo shows in a treatise entitled "Considerazione sopra il
giiico (lei dadi" that these combinations cannot be regarded as
being ecjually likely. By painting each of the three dice with
the diilVrent color it is easy to see that an arrangement such as
1 2 () can be produced in G different ways. Let the colors be
white, black and red resi)ectivcly. We may then make the
following arrangements:
White
Black
Red
1
2
6
1
G
2
2
1
G
2
G
1
6
1
2
6
2
1
which gives 3! = G different arrangements. The arrangements
of 1 4 4 can be made as follows:
White Bhick
Red
1 4
4
4 1
4
4 4
1
which gives 3 different arrangements. The arrangements of
3 3 3 can be made in one way only. By complete enumeration
of equally favorable cases we obtain the following scheme:
Sum 9
cases
Sum 10
cases
1, 2, G
G
1, 3, G
6
1, 3, 5
G
1, 4, 5
6
1,4,4
3
2,2,6
3
2, 2, 5
3
2, 3, 5
6
2, 3, 4
6
2,4,4
3
3, 3, 3
1
3,3,4
3
25
27
The total number of equally possible cases by the different ar-
rangements of the 18 faces on the dice is 6^ = 21G. The prob-
ability of throwing 9 with three dice is therefore gYc > ^^f throwing
■^" 2 16 8-
20 THE MATHEMATICAL THEORY OF PROBABILITIES [14
14. Example 2. — D'Alembert, the great French mathematician
and natural philosopher and one of the ablest thinkers of his
time, assigned | as the probability of throwing head at least
once in two successive throws with a homogeneous coin. D'Alem-
bert reasons as follows : If head appears first the game is
finished and a second throw is not necessary. He therefore gives
as equally possible cases (we denote head by II and tail by T) :
H, TH, TT, and determines thus the probability as |. Where
then is the error of D'Alembert? At first glance the chain of
reasoning seems perfect. There are altogether three possible
cases of which two are in favor of the event. But are the three
cases equally likely? To throw head in a single throw is evi-
dently not the same as to throw head in two successive throws.
D'Alembert has left out of consideration the fact that a double
throw is allowed. The following analysis shows all the equally
I)ossible cases which nuiy occur:
////, Iir, Til, TT.
Three of those cases favor the event. Hence we have:
P(E) = p =
We shall return to this problem at a later stage under the dis-
cussion of the law of large numbers.
The examples quoted have already shown that the enumer-
ation of the equally likely cases requires a sharp distinction
between the different combinations and arrangements of ele-
ments. In other words, the solution of the ])roblems requires
a knowledge of permutations and combinations. We assume
here that the reader is already acquainted with the elements and
formulas from the combinatorial analysis and shall therefore
proceed with some more illustrations. In the following, when
employing the binomial coefficients, we shall use the notation
y!) instead of "'Cfc.
15. Example 3.- An urn contains a white and h black balls. A
person draws /.• balls. What is the probability of drawing a
white and /3 black balls?
(a + /3 = /.-, a < a, ^ ^ h)
1 .") ] EXAMPLE 3. 21
k balls miiy hv drawn from the urn in as many ways as it is possible
to select k elements from a + b elements, which may be done in
((i+h\ _ (a+b\
ways. Furtliermorc there are I I ^roui)s of a white and II
groups of j3 black balls. Since each combination of any one
<2;roui) of the first groups with any one f^rouj) of the second groups
is favorable for the ?vent, we have as favorable cases:
Example 4. A special case of the above problem is the fol-
lowing: (jucstion which often appears in the well known game of
whist. What are the respecti\e chances that 0, 1, 2, 3, 4 aces
are held by a specified player? There are altogether 52 cards
in the game equally distributed among 4 players. Of these
cards 4 are aces and 4S are non-aces. Hence we have the fol-
loAving values for a, b, Ic, a and /3.
a ^ A, b = 4S, /.• = U, a = 0, 1, 2, .3, 4, /3 = 13, 12, 11, 10, 0.
Substituting in the abo^•e formula we get:
/4\ /4S\ /o2\ S2251
P«=(())Xll3)^ll3) = 270725
?^^=\l)Xll2) --113; = 270725
/4\ /4Sv /:)2\ 57798
^^^ = I2) X 111) ^ll3) = 270725
/4\ /4SV /52\ 11154
^^=l3)><llo)^ll3) = 270725
?^^=^ (4) ><(<)) -^(13) = 270^-
A hypothetical disjvuicti\e judgment immediately tells us that in
22 THE MATHEMATICAL THEORY OF PROBABILITIES. [ 16
a game of whist a specified player must either hold 0, 1, 2, o or
4 aces. Any such judgment is certain to come true. Hence by
adding the 5 above computed probabilities we obtain a check
for the accuracy of our calculations. The actual addition of the
numerical values of /;,), [h, p-i, P:i, and p^ gives us unity which is
the mathematical symbol for certainty. Gauss, the renowned
German mathematician and astronomer, was an eager whist
player. I )uring his f orty-eigiit years of residence in the university
town of Gottingen almost every evening he played a rubber of
whist with some friends among the university professors. He
kept a careful record of the distribution of the aces in each
game. After his death these records were found among his
papers, headed "Aces in Whist." The actual records agree
with the results computed above.
16. Example 5. — An urn contains n similar balls. A part of
or all the balls are drawn. What is the probability of drawing
an even number of balls?
One ball may be drawn in as many ways as there are balls,
two balls in as many ways as we may select two elements out of
72 elements, and so on. Hence we have for the total number of
equally possible cases:
-(;v(:)+(3)+-+(+')"(:)-
We have now:
and
. a-.ir = i-(;v(:;)-...+(-i)»(;;).
The number of favorable cases is given by the expansion:
/=(:)+(:)+
The expression for / is the binomiiuil coefficients less unity.
Hence we have:
t = (] -\r 1)" - 1 = 2" - 1.
If we add the two expansions of (1 + 1)" and (1 — 1)" and then
17] EXAMPLE 6. 23
subtract 2 we get tlic expansion for 2/. Hence we have:
2/ = [(1 + 1)" + (1 - D" - 2] .-. / = 2»-' - 1.
Thus we shall have as the probability of drawing an even numi)er
of balls:
2«-i _ 1
^^^^^-^^
while for an uneven number:
_ ^ = ^
9n— 1
We notice that the probability of drawing an uneven number of
balls is larger than the probability of drawing an even number.
This apparently strange result is easily explained without the
aid of algebra from the fact that when the urn contains one ball
only, we cannot draw an even number. Hence we have p = 0,
q = \. With two balls we may draw an uneven number in two
ways and an even number in one way, thus p = \, and q = §.
The greater weight of q remains when n is finite; only when
n = CO, P = q = 2-
17. Example 6. — A box contains n balls marked 1, 2, 3, • • • n.
A person draws n balls in succession and none of the balls thus
drawn is put back in the urn. Each drawing is consecutively
marked 1, 2, 3, • • • n on n cards. What is the probability that
no ball marked a (a = 1, 2, 3, • • • n) appears simultaneously
with a drawing card marked a?
The number of equally possible cases is simply the number of
permutations of n elements which is equal to nl
The number of favorable cases is given by the total number
of derangements or relative permutations of n elements, i. e.,
such permutations wherein the numbers from 1 to n do not appear
in their natural places. The formula for such relative permuta-
tions was first given by Euler in a memoir of the St. Petersburg
Academy entitled "Quaestio Curiosa ex Doctrina Combina-
tionis." Euler makes use of a recursion formula. A German
mathematician, Lampe, has, however, derived the formula in a
simpler manner in "Grunert's Archives" for 1884.
24 THE MATHEMATICAL THEORY OF PROBABILITIES. [17
Lampe denotes by the symbol ^(1) the number of permuta-
tions wherein 1 does not appear in its natural place. By letting 1
remain fixed in the first place we obtain (n — 1) ! permutations of
the other remaining elements, or:
<^(1)„= nl- {u- 1)1
permutations where 1 is out of place. Of these permutations
there are, however, a number wherein 2 appears in its natural
place. If we let 2 remain fixed in this place we shall have:
<^(l)„_i= {n- l)\- {n-2)\
permutations wherein 2 is in its place but 1 out of place, there
remains thus:
<pi2)„ = <^(1)„ - <^(l)„_i = u\- 2(n -1)1+ (n- 2)1
permutations in which neither 1 nor 2 is in its natural place.
Letting 3 remain fixed in its place, the remaining n — 1 elements
give:
(n- 1)!- 2{n- 2)!+ (n - 3)!
cases where 3 is in its place but 1 and 2 are not. Accordingly
there will be:
,^(.3)„ = <p{2)n - <pi2)n-x =nl- 3(« - 1)! + 3(n - 2)! - (/7-3)!
permutations in which none of the three elements 1, 2, and 3 is
in its place. The complete deduction gives us now for the
number r:
<p{r)n = n\ - ([) {n - 1)! + [[) {n -2)1
+ (- 1)' (I) in - r)l
arrangements in which none of the numbers 1, 2, 3, • • • r is in
its place. Hence the reciuired probability is:
'^^=i-cy- + r]^^
nl \1/ // \2/ „(n - 1)
"^ ^ ^^\r)n(n- 1) • • • (n -
(n - 1) • • • (n - r + 1) '
17 ] EXAMPLE 6. 25
when n = r the above expression becomes:
or the probability that none of the balls appear in its numerical
order.
When /? = oc the above ex])ression converjijes towards e~^ as
a limit. Since the series is rapidly convergent, we may therefore
as an approximate \'alue let
jj = t'-i = 0.36788 • • • .
The probability that at least one ball appears in numerical order
is
q= I- p = 0.G3213 •••.
CHAPTER TV.
THE ADDITION AND MULTIPLICATION THEOREMS IN
PROBABILITIES.
18. Systematic Treatment by Laplace. — The reader will readily
have noticed that the problems hitherto considered have been
solved by a direct application of the fundamental definition of a
mathematical probability. Almost every branch of pure and
applied mathematics has originated in this manner. A few
isolated problems, apparently having no mutual connection what-
soever, have presented themselves to different mathematicians.
As the number of problems increased, there was found to exist
a certain inner relation between them, and from the mere isolated
cases there grew a systematic treatment of an entirely new
subject.
The theory of probabilities had its origin in games; and the
different problems that arose, were treated individually. From
the time of Galileo and Cardano to the appearance of Laplace's
great treatise, a number of celebrated mathematicians such as
Pascal, Fermat, Huyghens, De Moivre, Stirling, Bernoulli and
others had solved numerous problems, some of these, as we already
have seen in the preceding chapter, of a quite complex nature.
But none of these mathematicians had liitherto succeeded in
giving a systematic treatment of the subject as a whole. All
their treatises were, as any one taking the trouble to look over
the works of De Moivre and Bernoulli will readily notice, mere
collections of examples solved by direct ai)plication of om- funda-
mental definition. It remained for Laplace first to give the
definite rnles to the science by which the solution of a great
number of problems, often very c()m])licate(l, was reduced to
the application of a few stable principles, first given in his
"Theorie Analytique des Probabilites " (Paris, 1812).
19. Definition of Technical Terms. — Before entering into a
demonstration of Laplace's theorems it will, however, be neces-
sary to explain a few technical terms which seem commonplace
26
20] THEOREM OF COMPLETE OR TOTAL PROBABILITY. 27
and simple enough but which, nevertheless, must be defined
clearly in order to avoid any ambiguity.
In all works on probabilities when speaking of happenings of
various events we encounter often the terms, indcpcndoit events,
dependent events and mutually exclusive events. An e\'ent E is
said to be independent of another event F when the actual
happening of F does not influence in any degree whatsoever the
probability of the happening of E. On the other hand, if the
probability of E is dependent on or influenced by the previous
happening of F, then E is said to be dependent on F. Finally the
two events E and F are said to be mutually exclusive when
through the occurrence of one of them, say F, the other event
E cannot take place, or vice versa. We might also in this case
consider the two events E and F as members of a complete dis-
junction. In a complete hypothetical disjunctive judgment as
"When a die is thrown either 1, 2, 3, -i, 5 or G will turn up"
each member represents a possible event. Any one of these
events is mutually exclusive in respect to the other events of the
disjunction.
20. The Theorem of the Complete or Total Probability, or the
Probability of " Either Or." — When an event, E, may happen in
any one of the /( different and mutually exclusive ways Ei, E^,
Ez, • • ■ En with the respective probabilities: pi, p-i, i:)^, • • • p^,
then the probability for the happening of the event, E, is equal
to the sum of the individual probabilities: pi, p^, pz, ■ • • pn-
Proof: The main event, E, falls in n groups of subsidiary events
of which only one can happen in a single trial but of which any
one will bring forth the event E. Let us by t denote the total
number of equally possible cases. Of these possible cases / are
in favor of the event. This favorable group of cases may now
be divided into n sub-groups of which /, are favorable for the
happening of E\, f-i in favor of £'2, fz in favor of £3 • • • /„ in
favor of En. When we write:
P(E) = v = j= ^ =7 + 7
-l-^+ ... +-^.
^ t ^ ^ t
28 THE ADDITION AND MULTIPLICATION THEOREMS. [21
Each of the fractions f^'t (a = 1, 2, 3, • • • n) represents the
respective probabilities for the actual occurrence of the subsidiary
events, Ei, E-i, Ez, • • • En- Hence we shall have
P{E) ^ p = pi + P2 + />3 + • • • + Pn.
This theorem is also known as the Addition Theorem of proba-
bilities. Instead of "total probability" the German scholar,
Reuschle, has suggested the expressive name of the "either or"
probability. The term is well selected when we remember that
the event, E, will happen when either Ei, or E^ or Ez • • • or En
happens.
Example 7. — What is the probability to throw 8 with two dice
in a single throw?
The total number of ways is t = Or = 36. The event in. ques-
tion E is comj)osed of the three subsidiary events favoring the
combination of 8:
E,: G, 2
£2: 5, 3
Ez: 4, 4.
Now
/
^(^') = i = i^' ^(^') = i = i^' ^(^') = li
Hence
^ ' 18^18"^ 36 36'
21. Theorem of the Compound Probability or the Probability
of " As Well As." — An event E may ha])pen when every one of
the mutually exclusive events Ei, E-i, Ez, • ' • En has occurred
previously. It is immaterial if the n subsidiary events have
happened simultaneously or in succession. But it makes a
difference if the events Ei, E2, Ez, • • • En are independent, or
dependent on each other.
1. Independent Events. — The probability, P{E) = p, for the
simultaneous or consecutive appearance of several nnitually ex-
clusive events: Ei, E>, ■ • • /?„ is equal to the product: pi-p-i-pz-
• • • p„ of the individual probabilities of the n events.
Proof: Let the number of possible cases entering into the
complex that brings forth the event E be /. Each of the t^
21 ] THEOREM OF THE COMPOUND PROBABILITY. 29
possible cases corresi)()nding to the event Ei may occur simul-
taneously with each one of the t'l cases corresponding^ to the event
Eo- Thus we have altogether /i X /•> cases falling on A'l and E2
at the same time. Continuing in the same way of reasoning it
is readily seen that the total number of equally possible cases
resulting from the simultaneous occurrence of the events Ei, Eo,
Ez, • • -En is equal to /i X /'^ X /s X • • • tn. By applying the same
reasoning to the favorable cases we get as their total number:
/ = /iX/. X/3X •••/„.
Hence the fiiuil probaliility for the hai)pening of the simultaneous
or consecutive appearance of the n minor events is:
P(£:) ={ = 7X7X7 X ••••-7= /^ix }hXp,x ••■vn.
Example 8. — A card is drawn from a whist deck, another card
is drawn from a pinochle deck. What is the probability that
they both are aces?
A whist deck contains o2 cards of which four are aces, a
pinochle deck 48 cards with 8 aces. Denoting the probabilities
of getting an ace from the whist and j)inochle decks by P(Ei)
and P{,E2) respectively we have:
P{E) = P{E,)P{E,) = ^^X^=~.
2. Dependent Events. — The n events £1, Eo, E3, • • • En are
not independent of each other, but are related in such a way that
the appearance of Ei influences E2, that event influences in turn
£3, Ei event £4 and so on.
The same reason holds as above, and,
P{E) = p = pi X p-i X p,X •■■ Pn.
But p-i means here the probability for the happening of E-i after
the actual occurrence of Ei, ps the probability for the happening
of E3 after Ei and E2 have pieviously happened, and so on for
all n events.
Example 9. — A card is drawn from a whist deck and replaced
by a joker, and then a second card is drawn. What is the prob-
abilitv that both cards are aces?
30 THE ADDITION AND MULTIPLICATION THEOREMS. [22
Denoting the two subsidiary events by Ei and E2 we have:
P{E) = P(£i)P(E.>)
4
3 3 3
52
52 13 X 52 676
The two above theorems are known as the multiphcation theorems
in probabihties. Reuschle lias also suggested the name " the
as well as ])r()bal)ility."
22. Poincare's Proof of the Addition and Multiplication
Theoreiti. — The French mathematician and physicist, II.
Poincare, has derived the above theorems in a new and elegant
manner in his excellent little treatise: " Lecons sur le Calcul des
Probabilites," Paris, 1896.
Poincare's proof is briefly as follows:
Let El and E2 be two arbitrary events.
El and E2 may happen in a difi'erent ways.
El may happen but not E2 in /3 different ways.
E2 may happen but not Ei in 7 different ways.
Neither Ei nor E2 will happen in 5 different ways.
We assume the total a + j8 + 7 + 5 cases to be equally likely to
occur.
The probability for the occurrence of Ei is
^' a + ^ + y + 8'
The probability for the occurrence of E2 is
a + 7
7>2
«+^+7+5
The probability for the occurrence of at least one of the events Ei
and E2 is
a + / 3 + 7
^' a + ^ + 7 + 5-
The probability for the occurrence of both Ei and E2 is
a
Pa =
« + |8 + 7 + 5'
The probability for the occurrence of Ei when E2 has already oc-
curred is
a
Pb = , .
a + 7
23 ] RELATIVE PROBABILITIES. 31
The probability for the occurrence of Eo when Ei has already oc-
curred is
a
The probability for the occurrence of Ei when E2 has not already
occurred is
^' = ^ + y'
The probability for the occurrence of E2 when Ei has not already
occurred is
y
7 +
We have now the following identical relations:
Vi + P2 = P3 + Pa, Pz = P\ + P2 — Pi,
i. e., the probability that of two arbitrary events at least one
will happen is equal to the probability that the first will happen
plus the probability that the second will happen less the prob-
ability that both will happen. The particular problem which
we may happen to investigate may possibly be of such a nature
that the two events Ei and E^. cannot happen at the same time,
in that case pi = 0, and we get:
Pz = i^i + Pi.
In this equation we immediately recognize the addition theorem
for two mutually exclusi\-e events. By substitution of the
proper values we have furthermore:
Pi = p-2 ■ Pb or Pi = pi • Pq.
These equations contain the theorems proved under §21, of
the probability for two mutually dependent events,
23. Relative Probabilities. — We shall now finally give an alter-
native demonstration of the same two theorems. It will, of
course, be of benefit to the student to see the subject from as
many view points as possible; moreovei;, the following remarks
will contain some very useful hints for the solution of more com-
plicated problems by the application of so-called " relative prob-
32 THE ADDITION AND MULTIPLICATION THEOREMS. [23
abilities "and a few elementary theorems from the calculus of
lojjic. The following paragraphs are mainly based upon a
treatise in the Proceedings of the Royal Academy of Saxony,
by the German mathematician and actuary, F. Ilausdorff.
In our fundamental definition of a mathematical probability
for the ha])peninp; of an e\-ent E, expressed in symbols by P(E),
as the ratio of the equally favorable and equally possible cases
resulting from a general complex of causes, we were able to
compute the so-called ordinary or absolute probabilities. But
if we, from among the favorable cases and possible cases, select
only such as bring forward a certain different event, say F, then
we obtam the " relative probability " for the happening of E
under the assumption that the subsidiary event, F, has occurred
previously. For this relative probability we shall employ the
symbol Pf{E), which reads "the relative probability of E,
positi F." The following problem illustrates the meaning of
relative probabilities. If an honor card is drawn from an
ordinary deck of cards, what is the probalMlity that it is a king?
Denoting the subsidiary event of drawing an honor card by F,
and the main event of drawing a king by E, we may write the
above mentioned probability in the symbolic form: Pp{E). If
on the other hand we knew a priori that a king was drawn, we
may also ask for the probability of having drawn an honor card.
Since any king also is an honor card, we may write in symbols:
Pe{F) = 1.
Before entering upon the immediate determination of relative
probabilities we shall first define a few symbols from the calculus
of logic. We denote first of all the occurrence of an event E
by E, the non-occurrence of the same event by E. Similarly
we have for the occurrence and non-occurrence of other events,
F, G, 11, • ■ ■ and /', d, II, • ■. E -Jr F means that at least one
of the two events E and F will happen. E X /' or simply E • F
means the occurrence of both E and F. From the above
definition it follows immediately that E + F = E • F and
/:=/•:• F + /•; • f.
This last relation simply states that E will ha])|)(Mi when either
E and /' happen simultaneously or when E and the non-aj^pear-
ance of /' happen at the same time. If furthermore Fu Fi, Fo,
25] PROBABILITY OP^ Rp:i'ETnTONS. 33
Fi ■ • • Fn, Fn constitute the inciiihers of a complete disjunetion,
i. e., mutually e\chisi\'e events, we have in <jeneral:
;•; = E ■ F, + /•: ■ I\ + /•; • f, + /•; ■F2+---F- f„ + /:: • f „.
From the orii^inal definition of a ])rol)abilit\ , it follows now:
P{K) = P(E ■ F) + PiE -F), ^
and
P{E) = P{E • F,) + PiE • F,) + P{E • /',) + P{E ■ F,)
+ PiE ■ Fn) + P{E ■ ¥\),
i. e., the probability that of several mutuallx' exclusive events
one at least will hai)])en is the sum of the probabilities of the
happening of the separate events. This is the symbolic form for
the addition theorem.
24. Multiplication Theorem. — We next take two arbitrary
events. From these events we ma\' form the following com-
binations:
E ■ F, E ■ F, E ■ F, E ■ F, I e.,
Both E and F happen,
E ha])])ens but not /'
F happens but not E
Neither E nor F happens.
Furthermore let a, (3, y, 5, be the respective numbers of the
favorable cases for the al)ove four combinations of the events
E and F. Following the previous method of Poincare, we shall
have:
a -\- y a -\- p
PiE ■ F) =
a + /3 + 7 + 5
25. Probability of Repetitions. — From the above equations it
immediately follows:
PiE ■ F) = PiE) X P,:iF) = PiF) X PriE),
which is the symbolic form for the multiplication theorems of
compound probabilities.
4
34 THE ADDITION AND MULTIPLICATION THEOREMS. [25
In special cases it may ha])])en that the different subsidiary
events: Ei, E-i, E3 ■ ■ • E„ are all similar. We shall then have,
following the symbolic method:
E = E, - E, ■ E^ - ■ • En = E, ■ E, ■ E, ■ ■ ■ E, = E,-,
and
P{E) = P(£i") = P(E,r.
This gives us the following theorem:
The probability for the repetition n times of a certain event,
E, is equal to the ??tli power of its absolute ])robability.
Thus if P{E) = p we have immediately P{E) = 1 — p.
PiE") = PiEY = //'.
P{E") = P{Er = (1 - pY-
Thus the probability for the occurrence of E at least once in
n trials is •
P{E + E + ■•■ n times) = 1 - P{E^) = 1 ^ (1 - p)".
Denoting the numerical quantity of this probability by Q we
have:
1 _ (^ = (1 _ p)n.
Solving this equation for n we shall have:
log (1-0
n =
log (1 - p)
Whenever n equals, or is greater than, the above logarithmic
value for given values of Q and p we are sure that Q will exceed
a previously given proper fraction. To illustrate:
Example 10. — How often must a die be thrown so that the
probability that a six appears at least once is greater than ^?
Here p = \, Q = 2- Hence we must select for n the smallest
positive integer satisfying the relation:
log i\-\) ^ !2^ _ .301035 . ^
log (1 - i) log I .079186 '■ ^^ "" ^'
For this particular value of n we have in reality:
Q = 1 - & = -318.
27] EXAMPLE 12. 35
26. Application of the Addition and Multiplication Theorems
in Problems in Probabilities. — We shall next proceed to illustrate
the theorems of the preceding paragraphs by a few examples.
First, we shall ap})l}- the demonstrated theorems to some of the
examples we have already solved by a direct a{)plication of the
fundamental definition of a inathcniatica! !)i-()l)al)ility.
Example 11. — We take first of all our old friend, the problem
of D'Aleinbert. What is the probability of throwing head at
least once in two successive throws with an uniform coin?
This j)robleni is most easily sohcd by finding the i)robability
first for not getting head in two successive throws. By the
multii)lication theorem this probability is: p = | X | = j.
Then the probability to get head at least once is 1 — j = f
from a simi)le application of the rule in § 25. A more lengthy
analysis is as follows. Denoting the event by E, the following
cases may appear which may bring forth the desired event:
Head in first throw which we shall denote by //i and head in
second throw wliich we denote by //.., or head in first throw {II x)
and tail in second (7'.), or finally tail in first {Ti) and head in
second (7/2). Then we have:
E= Ih- II, + Ih ■ T,+ T,- Ih,
or:
P{E) = P{H,) . P{Ih) + P{H,) ■ P(T,) + P{T,) . P(H,)
— ivi-i-ivi_i_ivi — 3
27. Example 12. — "\Miat is the probability of throwing at
least twelve in a single throw with three dice? The expected
event occurs when either 12, 13, 14, . . . or 18 is thrown. Of
these events only one may happen at a time. We may, there-
fore, apply the addition theorem and obtain as the total prob-
ability :
p = pn + Pn + Pi4 + • • • + pis.
where pn, Pis, • • • Pm are the respective probabilities for throwing
the sums of 12, 13, ••• or IS. These subsidiary probabilities
were determined in § 1.3 under the problem of Galileo, and:
T) = ^2_6_ I JUL J ^1_5_ 4. JUL _4_,fi__l__^ I 1 _27
y 216^2161216^^216^^16^^ 2T6 "T 216 " ^2-
36 THE ADDITION AND MULTIPLICATION THEOREMS. [ 2 >^
28. Example 13. — An urn contains a white, h black and c red
balls. A sin<:jle ball is drawn a + /3 + 7 times in succession,
and the ball thus drawn is replaced before the next drawing takes
place. To determine the probability that (1) there are first
a white, then 13 black and finally 7 red balls, (2) the drawn balls
appear in three closed groui)s of a white, 13 black nnd 7 icd balls,
but the order of these groups is arbitrary, (3) that white, black
and red balls appear in the same number as above, but in any
order whatsoever.
1. Denoting the three subsidiary events for drawing a white,
|3 black and 7 red balls by Fu F-i and Fz, and the main event for
drawing the balls in the prescribed order by E, we may write the
probability for the occurrence of the main event in following
symbolic form involving symbolic probabilities:
P{E) = P{F{)Pj.,{F,)P^,^.S.Fz).
Substituting the algebraic values for P{Fi), PiF^) and P{Fz)
in the expression for P{E), and then. applying Hausdorff's rule
(§24) we get:
g" h^ c*
P{E) - Pi - ^^ _^ ^ _,_ ^)a X ^^ _^ ^ _^ ^)3 X ^^ _^ ^ ^ ^^,
(a+6 + c)''+^+^'
2. In the second part of the problem the order of the three
different groups is immaterial. The three subsidiary events:
Fi, Fi and F^, may therefore be arranged in any order whatsoever.
The total number of arrangements is 3! = 6. The probability
of the happening of any one of these arrangements separately
is the same as the probability comi)uted luider (1). By ai)})lying
the addition theorem we get therefore as the probability of the
occurrence of this event:
6 a''6^cy
^'' ~ (a+~6 + c)»-"^+^*
3. The third part is more easily solved by a direct application
of the definition of a mathematical probability. Th(> order of
the balls drawn is liere immaterial. Of each individual com-
30] EXAMPLE 15. 37
bination of a white, jS black balls and 7 red balls it is possible
to form (a + /3 + y)\la\&\y\ different i)ennutations as the total
number of faA'orable cases. The above number of equally j)os-
sible cases is here {a -\- h -\- cY^^^'^. Hence we have:
(a + /? + 7) ! a'^h^c*
Vz = TVoTTl X
a 1/3 17! ^^ (a+ fe + c)"+^+^"
29. Example 14. — In an urn are n l)alls amon<; which are a
white and ^i black. What is the prt)bability in three successive
drawings to draw (1) first two white and then one black ball, (2)
two white and one black ball in any order whatsoever? (a+jS^w).
The probability to draw first one white, then another white and
finally a black ball is:
^ « (« - 1) _J^__
^^ n{n - 1) {n - 2)'
The probability for any of the other arrangements is the same,
or we have for (2)
3a {a-\) ^
j)2 = 3pi = — X
n {n - 1) (w - 2) •
30. Example 15.— What is the chance to throw a doublet of
6 at least once in n consecutive throws with two dice? (Pascal's
Problem.)
Chevalier de ]\Iere, a French nobleman and a great friend of all
games of chance, went more deeply into the complex of causes in
different games than most of the ordinary gamblers of his time.
Although not a proficient mathematician he understood suffi-
cient, nevertheless, to give some very interesting problems for
which he got the ideas from the gambling resorts he frequented.
De ^Nlere was a friend of the great French mathematician and
philosopher, Blaise Pascal, and went to him whenever he w^anted
information on some apparently obscure point in the different
games in which he participated. The chevalier had from patient
observation noticed that he could profitably bet to throw a six
at least once in four throws witii a single die. He reasoned now
that the number of throws to throw a doublet at least once with
two dice ought to be proportional to the corresponding equal
number of possible cases with a single die. For one die there are
38 THE ADDITION AND MULTIPLICATION THEOREMS. [31
6 possible cases, for two 36. Thus de iNIere thought he could
s./icly bet to throw a doublet of 6 in 24 throws with two dice.
An actual trial by several games of dice proved extremely
disastrous to the finances of the nobleman, who then went to
Pascal for an explanation. Pascal solved the problem by a direct
application of the definition of a mathematical probability. We
shall, however, solve it by an application of the multiplication
theorem.
The probability to get a doublet of G in a single throw is -gQ.
The probability of not getting a double six is therefore 1 — ^q
= 11 . The probability of the ha])pening of this event n con-
secutive times is (f g)". Thus the probability of getting a double
six at least once in Ji throws with two dice becomes: p = 1 —
(ft)"- Solving this equation for n we shall have:
^ log (1 - p)
log 35 — log 30 '
for p = 2 ^'6 shall have:
log 2
n = = 24. G • • •.
log 3G — log 35 " '
First for 25 throws we may bet safely one to one while for 24
throws such a bet was unfavorable. This shows the fallacy of
de Mere's reasoning.
31. Example 16. — An urn,.l, contains a balls of winch a are
white, another similar urn, B, contains b balls of which /3 are
white. A single ball is drawn from one of the two urns. What
is the probability that the ball is white? The beginner may easily
make the following error in the solution of this problem. The
probability to get a white ball from A is a/a, from B, fi/b. Thus
the total probability to get a white ball is: a /a + j3,'b. This
result is, liowe\er, wrong, for we may, by selecting proper values
for a, b, a and /3, obtain a total probability which in numerical
value is greater than unity. Thus if a = 7, b = 7, a = 5,
/3 = 4, we get as the total probability:
P — Y T- 1 — y.
This result is evidently wrong, since a mathematical probability
is never an improper fraction. The error lies in the fact that we
32] EXAMPLE 17. 39
have regarded the two events of drawing a ball from either urn
as independent and mutually exclusive. A simple ai)j)lication
of the symbolic rule for relative probabilities will give us the
result immediately. The main event, K, is com])osed of the two
following subsidiary events: (1) to get a white ball from A, or
(2) to get a white ball from B. We shall symbolically denote
these two events by A ■ IT' and B ■ JV respectively. Thus we
have :
P{E) = P(A ■ JV) + P(B ■ Jn = P(A)PJ]V) + P(B)PJ\V).
Now the probability to obtain urn A is P{A) = pi = h, also to
get B: P{B) = p-y — h. The probability to get a white ball
from A when this particular urn is previously selected is expressed
by the relative probability:
Similarlv for B:
p.m = P^=l-
PB(in - pa = 1
Substituting these different values in the expression for P(E)
we get finallv:
For the particular numerical example we have:
1 /.") 4\ 9
H=+=)
P 2\7 ' 7/ 14
32. Example 17. — The probability of the happening of a
certain event, E, is p, while the probability for the non-occurrence
of the same event is </ = 1 — p. The trial is now to be repeated
n times. The probability that there will be first a successes
and then (3 failures is:
P{E^)Pe'^ (E^) = p" . (j^(a + (3= n).
This is the probability that the two complementary events E and
E happen in the order prescribed above. When the order, in
which the successes and failures happen, plays no role during
the n trials, that is to say it is only required to obtain a successes
40 THE ADDITION AND M I LTIPLICATION THEOREMS. [33
and i5 failures in any order whatsoever in n total trials, then the
arrangement of the a factors p and /3 factors q is immaterial. The
total number of arrangements of n elements of which a are equal
to p and 13 equal to q is simply nil (a ! X /? I) . For any one particu-
lar arrangement of a factors p and /3 factors q the ])rol)ability of
the ha])pening of the two complementary events in this particular
arrangement is equal to p"^ ■ q^ . The Addition Theorem im-
mediately gives the answer for a successes and /3 failures in any
order whatseover as:
l\E^-m = p^^{^l)pY
Let us, for the present, regard this probability as being a function
of the variable quantity, a, {n being a constant quantity). We
may then write:
p^ = ip{a).
Letting a assume all positive integral values from to n the
above expression for p^ becomes:
Po
P2
Pn = P
These are the respective probabilities for no successes, one success,
two successes, . . . and finally n successes in n trials. The
above quantities are, however, merely the diflFerent members of
the binomial expansion (p + q)". Since p -\- q = 1 from the
nature of the problem, we also have (p -\- q)" = 1, or po + pi
+ P2 + • • • -\- Pn = 1. This last equation is the symbolic
form for the simple hypothetical disjunctive judgment: E must
happen either 0, 1,2, • • • or 71 times in n total trials. We shall
return to this problem later under the discussion of the Bernoullian
Theorem. In fact, the above example constitutes an essential
part of this famous theorem which has proven one of the most
important and far reaching in the whole theory of probability.
33. Example 18. De Moivre's Problem. — The following prob-
lem was first given by the eminent French-English mathemati-
33] EXAMPLE 18. 41
cian, Abraham de ]\Ioivre, in a treatise, entitled " De ]\Iensura
Sortis," which was pubHshed in London about 1711.
An urn contains n -{- I balls marked 0, 1,2, • • • n. A person
makes i drawings in succession, and each ball is put back in the
urn before the next drawing takes place. What is the probability
that the sum of the numbers on the n balls thus drawn equals s?
The first ball may be drawn in n + 1 ways, the second ball
may also be drawn in w + 1 ways. Hence two balls may be
drawn in (n + 1)^ ways or i balls in (n + 1)' ways: This is the
total number of equally possible cases.
If we expand the expression:
(.ro + ,;•' + .r2 + .r3 + .r^ + ... .r'')' (1)
after the multinominal theorem, we notice that the coefficient
to x' arises out of the different ways in which 0, 1, 2, 3, • • • n
can be grouped together so as to form s by addition, which also
is the total number of favorable cases. The expression (1)
inside the bracket represents a geometrical progression, which
may be written as:
(1 - .r"+i)'(l - .r)-^ = I 1 - ^'•^•""^' + (i)-^'"'^' ~ (o) •1-^"+^
+ -.jx(i + «-.('-V)..=+Ct^'),.+ ...j.
By actual multiplication we get a power series in .r. The terms
containing .r^ are obtained in the following manner: the first term
of the first factor being multiplied with the term
(i -\- s — 1 \
I .r' of the second factor,
the second term of the first factor multiplied with the term :
(i -\- s — )i — 2 \
, I .j-*~"~^ of the second factor,
s — n — 1 f '
the third term of the first factor multiplied with the term:
I _ o ~_ o ) .T^-""" of the second factor.
42 THE ADDITION AND MULTIPLICATION THEOREMS. [34
Thus the coefficient of .r" is equal to
+ (;)r!:o;";^)--.
The above expression may by further rechictions be brought to
the form:
(^+l)(^+2) ■'■ (s+ i-1) -
1 • 2 • . . (i - 1)
i \ (,'i — n){s — n -\- I) • ' ■ (s — n -}- i — 2)
-{[)
+(;)
1 • 2 • • • (i" - 1)
•/ \ {s - 2n - l){s- 2n) ■■■ (s - 2n + i - 3)
1 • 2 ••■(/- 1)
The series breaks of course as soon as iiei^ativc factors appear
in the numerator. The required probabihty is therefore
. , (^+l)(^+2) ■■■ (s+i-l)
+
(»+ DM 1 • 2 ••• (i- 1)
(s — w)(.? — n + 1) • • • (s — n -\- i — 2)
-(;)
1, • 2 • • • (/ - 1)
34. Example 19. — If a single experiment or observation is
made on n pairs of opposite (comi)lementary) events, E^ and E^
with the respective probabilities of happening p^ and q^{a = 1,
2, o, • • • n), to determine the i)robability that: (1) exactly r,
(2) at least /• of the events E^ will happen.
This ])r()blem is of great ini])()rtan('e, especially in life assurance
mathematics. It happens frecpiently that an actuary is called
upon to determine the probability that exactly /• jiersons will be
alive m years from now out of a group of n persons of any age
whatsoever, each piM-son's age and his individual coefficient of
survival through the ixM-iod being known beforehand.
Various demonstrations have been given of this i)r<)blein. The
first elementary proof was probably due to ?^lr. George King,
34] EXAMPLE 19. 43
the English actuary, in liis well-known text-book. The Austrian
mathematician and actuary, E. Czuber, has simplified King's
method in his " Wahrscheinlichkeitsrechnung " (lOO^i). Later
the Italian actuary, Toja, has given an elegant proof in BoUctino
degli Aftitdri, WA. 12. Einally another Italian mathematician,
P. ]\Iedolaghi, has investigated the problem from the standpoint
of symbolic logic. In the following we siiall adhere to the demon-
stration of Czuber and also give a short outline of the symbolic
method.
In order to answer the first part of the problem we must form
all possible combinations of r factors of p and ?i — r factors of q
and then sum all such combinations of n factors. Denoting
the event by E r] 'vve have:
= -/^„7>3 • • • (1 - Pa)(1 - p,) •■• (1 - p„).
We shall now denote the sum of all products in (1) containing if
factors p by the symbol »S^. It is readily seen that ^ will have
all positive integral values from r to n inclusive. We may
therefore write the total compound probability in the following
form :
P{E^ri) = AoSr + ,llS.+ : -f .l.,S,+o + • • • + An-rSn. (2)
The student must bear in mind that the different S are merely
symbols for different s^nns of all the i)roducts of /•, r + 1, r + 2,
• • • n factors p respectively. Our problem is now to determine
the unknown coefficients A. It is easily seen that the coefficient
Aq = \, since all different products containing r factors p appear
only once. The other coefficients of the form A do not depend
on the values of p, however. They remain therefore unaltered
if we equate all of the various p's and let them equal p. Ex-
pression (1) then simi)ly becomes I j • p''{l — p)"~\ We must
form all possible rth powers of n similar factors, which can
be done in ( I ways. The expression (2) on the other hand
44 THE ADDITION AND MULTIPLICATION THEOREM. [34
becomes :
+ h An-rP''.
Any S^ is by definition the sum of all products containing (p
factors p and we may form I | sucii j^roducts from n elements
;;. But we saw above that <p might only have all positive values
from r to n inclusive, hence expression (2) will naturally take
tlic alxAe form. We have therefore
+ -'=(r+ 2) ■''"*'+ ■■■+-i„-,p\
Expanding the expression on the left hand side by means of the
binomial theorem and equating the coefficients of equal powers
of J), we get:
or:
Substituting these values in (2) for the unknown coefficients A,
we shall have:
P(E,r,) = Sr- (""l ) .SVl + ( ' 2 ) ''^'-'' ~
r+ 1\ ., , /r+ 2
34] EXAMPLE 19. 45
If we expand the algebraic expression:
= 5'-(l + 5)-('^-i)
:r+2
(1 + S)'^'
we have:
We may therefore write P(E) = v-TTT^y+i > when e\-ery exi)o-
nent is rej)hiced by an index number (i. c, .S* replaced by .S'^)
and the expansion broken off at the term *S'". The student must
of course constantly bear in mind the symbolic meaning of ^S^.
The second ])art of the problem is easily solved by the sym-
bolic method. Denoting this particular event hy Er, we have
the following identity:
P(Er) - P(Er+l) = P(E,r,)
or
P(Er) - PiE,r,) = P{Er+l).
The following relations are self-evident:
P{E,) =1;
P{Ei) = PiEo) - P(EJ - 1 - Y^, also;
P{E,) = P{E,) - P(£,„) = '^ ^ ^'
1+S (1 + 5)2 (1 + 5)2*
The complete induction gives us finally:
P(Er) .=
(1 +sy
Assuming the rule is true for /•, we may easily prove it is true
for r + 1 also. We have in fact:
(1 + sy (1 + s)^^ (1 + sy+^ '
46 THE ADDITION AND MULTIPLICATION THEOREMS. [35
35. Example 20. Tchebycheff's Problem. — The following solu-
tion of a very interesting problem is due to the eminent Russian
mathematician, Tchebycheff, one of the foremost of modern
analysts.
A proper fraction is chosen at random. What is the proba-
bility it is in its lowest terms?
Stated in a slightly different wording the same question may
also be put as follows: If A/B is a proper fraction, what is the
probability that .1 and B are prime to each other?
If p2, pz, Ph, • • • pm denote respectively the probabilities that
each of the primes 2, 3, 5, • • • m is not a common factor of
numerator and denominator of A/B, then the probability that
710 prime number is a common factor is:
P = p2 ■ pz ■ p;, ■ ■ • Pm ■ ■ ■ p, • • • ad. inf. (I)
This follows from the multiplication theorem and from the fact
that the sequence of prime numbers is infinite.
Tchebycheff now first finds the probability Qm = ^ — Pm that
the fraction A/B does contain the prime m as factor of both A
and B. By dividing any integral number by the prime m we
obtain besides the quotient a certain remainder that must be
one of the following numbers, viz.:
0, 1, 2, 3, 4, • • • (m - 1).
Each of the above remainders may be regarded as a possible
event. The probability to obtain as a remainder is accordingly
1/m. The probability that m is contained as a factor of .1 is
therefore 1/m. This same quantity is also the probability that
m is a factor of B. The probability that both A and B are
divisible by m is therefore:
1 111 .1
qm= I — Pm = = —^, or Pm = I — —i-
111 m m- m^
Hence we have for the various primes:
P2=l-r^, P3=l— 5^, P5=l-T^, ••••
35] EXAMPLE 20. tchebycheff's problem. 47
Formula (I) then takes the form:
Forming the reciprocal l/P we get:
1111
P l_i i_i i_l
22 3'-' 52
ad. inf.
Now each factor on the right hand side is the sum of a geometrical
progression, as:
p = ^ 1 + ^ + ^2J2 + • ■ • / \ 1 + 32 + (3^ + • • • f
( 1 + 55 + (^2 +•••)••• ad. inf.
Multiplying out we shall have :
^^^T2~i92~r'Q2'T2~'r2' '*' ^^' iril-
The above infinite series is, however, merely the well known
Eulerian expression for tt^/G, hence:
Suppose furthermore we were assured that none of the three
primes 2, 3, 5 was a common factor of both A and B. What
would then be the probability that the fraction might be reduced
by division by one or more of the other primes?
Denoting by the symbol Pa) the probabiHty that none of the
primes from 7 and upw' ards is a common factor, we get :
also:
^ = ^= (1-2^) (1-32) (1-52)^(7),
48 THE ADDITION AND MULTIPLICATION THEOREMS, [35
or:
p.. = ^[(.4)('-^)(-l)] = --.
The probability of the divisibihty of both numerator and
denominator of a fraction chosen at random by a prime hirger
than 5 is tlius:
J_
20
1 - P(7) = -.
The summation of the infinite series of the reciprocals of the
squares of the natural numbers baffled for a long time the skill
of some of the most eminent mathematicians. Jacob Bernoulli,
the renowned classical writer on probabilities, proved its conver-
gency but failed to find its sum. The final summation was first
performed by Euler.
CHAPTER V.
MATHEMATICAL EXPECTATION.
36. Definition, Mean Values. — It is common belief among
many people that gambling and all kinds of betting have their
source in reckless desire. This is often argued by moral reform-
ers, but cannot be said to be the true cause. Whenever by ordi-
nary gambling or by a bet, actual value is exposed to a complete or
partial loss, this exposure is not due to the fact that the gamester
is reckless, but because there is hope of an actual gain. "Hope,"
says Spinoza in his treatise on ethics, "is the indeterminate joy
caused by the conception of a future state of alTairs of whose
outcome we are in doubt." Actual mathematical calculation
cannot be attempted on the basis of this definition any more
than it could be attempted to determine a mathematical prob-
ability from the definition of Aristotle. "We disregard there-
fore the psychological element of desire, which is associated with
hope or expectation as well as the anxiousness or dread associated
with the related psychological element of non-desire" (Cantor).
The so-called niathrinafical expectation is the product of an
expected gain in actual \alue and the mathematical probability
of obtaining such a gain. The danger of loss may in this case
be regarded as a negative gain. Thus if a person, A, may expect
the gain, G, from the event, E, whose probability of happening
is equal to p, then e = p-G is his mathematical expectation.
The quantity expressed by the symbol, e, is here the amount it
is safe to hazard for the expected gain, G. We may also regard
the quantity, c, as a mean value or average value. Among a
large number of Ji cases only np will bring the gain, G, the others
not. Thus the total gain is:
pjiG -=- w = pG.
Suppose we have n mutually exclusive events, Ei, Ei, • • • , En,
5 49
50 MATHEMATICAL EXPECTATION. [36
forming a complete disjunction. For their respective prob-
abilities we have then the following equation:
Ih + Ih + />3 + • • • + Pn = 1.
If the actual occurrence of a certain one of these events, say,
E^, brings a gain of 6'„, then the total value of the mathematical
expectation of the n events is:
e = Ih- Ch + Pi • 62 + • • • + Pn ■ Gn = "^Pa ' 0^.
Since Zp^ = 1 this result may be written:
f X (pi + i>2 + • • • + Pn) = Ol-Pl + 6'2-i>2 + G3-pz + • ' -Gn-pn,
hence e may be regarded as the mean value of the different
quantities G^ with the weights p^ (a = 1, 2, 3, • • •, n).
Although we shall discuss the theory of mean values in a
following chapter a few preliminary remarks might not be out
of place here.
A variable quantity A' is related to a series of events Ei, Eo,
Ez, • • • , En (it being assumed that these events form a complete
disjunction) in such a manner that when E^ happens X takes on
the value .r^ (a = 1, 2, 3, • • •, n). If furthermore p\, />>, pz, • ■ ■
denote the respective probabilities of the occurrence of Ei, E2,
Ez, • • -, then
M{X) = piXi + 7>2.r2 + ■ • -Pn^-n
is called the mean value or simply the mean of X.
The above definition may be illustrated by the following
concrete urn-scheme. An urn contains N balls of which a\ balls
are marked .vi, a-i balls marked .12 • • • and finally a,, balls marked
Xn where Oi + ch + 03 + • • -On = X. Each drawing from the
urn produces a certain number A^, which may assume n different
values xu X2, xz, • • •, Xn, each with the respective probabilities:
V^ = N'^ = N-'-^- = X-
The arithmetic mean of all the numbers written on the balls is:
fll-Ti 4- ChXi + • • • anXn
N
which agrees with the mean as defined above.
38] VARIOUS EXPLANATIONS OF THE PARADOX. 51
37. The Petrograd (St. Petersburg) Problem.— In this con-
nection it is worthy to note a celebrated problem, which on
account of its paradoxical nature has become a veritable stumb-
ling block, and has l)een discussed by some of the most eminent
writers on probabilities. The problem was first suggested l)y
Daniel Bernoulli in a coniniuiiication to the Petrograd — or as
it was then called St. Petersburger Academy — in 173S.
The Petrograd problem may shortly be stated as follows: Two
persons A and B are interested in a game of tossing a coin under
the following conditions. An ordinary coin is tossed until head
turns up, whicii is the deciding e\ent. If head turns up the first
time A pays one dollar to B, if liead appears first at the second
toss B is to receive two dollars, if first at the tliird time four
dollars and so on. What is the mathematical expectation of /i?
Or in other words, how much must B pay to A before the game
starts in order that the game may be considered fair?
The mathematical expectation of B in the first trial is
^ X 1 = ^. The mathematical expectation for head in second
throw is (^)- X 2 = ^. Or in general the mathematical prob-
ability that head appears for the first time in the nth toss is
(§)", and the co-ordinated expectation is 2""^-^ 2" = ^. Thus the
total expectation is expressed by the following series:
1 J- 1 + 1 -I- ...
2 12 12 1
"\Mien n = oo as its limiting value it thus appears that B
could afford to pay an infinite amount of money for his expected
gain.
38. Various Explanations of the Paradox. The Moral Expec-
tation. — Tliis e\idently paradoxical result has called forth a num-
ber of explanations of various forms by some eminent mathe-
maticians. One of the commentators was D'Alembert. It was
to be expected that the famous encyclopaedist, who — as we have
seen — did not view the theory of probabilities in too kindly a
manner, would not hesitate to attack. He returns repeatedly
to this problem in the "Opuscules" (1761) and in " Doutes et
questions" (Amstenhiin, 1770).
D'Alembert distinguishes between two forms of possibilities,
viz., metaphysical and i)liysical possibilities. An event is by
52 MATHEMATICAL EXPECTATION [38
him called a metaphysical possibility, when it is not absurd.
When the event is not too "uncommon" in the ordinary course
of happenings it is a physical possibility. That head would
appear for the first time after 1,000 throws is metaphysically
possible but quite impossible physically. This contention is
rather bold. "What would," as Czuber remarks, "D'Alembert
have said to an actual rej)()rted case in 'Grunert's Archiv' where
in a game of whist each of the four players held 13 cards of one
suit." The numerical probability of such an event as expressed
by mathematical probabilities is (635013559600)-^.
D'Alembert's definitions including the half metaphorical term
"ordinary course" are rather \ague. And what numerical
value of the matheinaticid proi)ability constitutes the physical
impossibility y D'Alembert gives three arbitrary solutions for
the probability of getting head in the nth throw, namely :
1 _1_ 1
2»(1 + /3n2) ' 2"+*^" ' , 2^B '
2" +
' — rj^'?.^
(K-n)
where a, 13, B, K are constants and 7 an une\'en number.
Daniel Bernoulli himself gives a solution wherein he introduces
the term "moral expectation." If a person possesses a sum of
money equal to x then according to Bernoulli
kdx
dy =
X
is the moral expectation of .r, k being a constant quantity.
Integrating we get:
1 (h/ = k I — = A-(log b — log a) = k log- ,
which is the moral expectation of an increase h — a of an original
value a. If now x denotes the sum owned by />' we may replace
the mathematical expectation by their corresponding moral ex-
pectations, that is to say replace 2"-V2" by (1/2") log ((a + 2''-i)/.r)
and we then have:
idi ^+^.11 ^+2, 1. .r+2''\
^\^^log-^+ ilog— ^-h •••^.log ^ j,
38] EXPLANATION OF PARADOX. 53
wliich is a coiver^ent series. In this connection, it may be
mentioned that the Bernoullian liypothesis has found (juite an
extensive use in tlie moch-rn theory of utihty.
De Morgan in his spK'iidid httle treatise "On ProbahiHties"
takes the \'ie\v that the solution as first given is by no means an
anomaly. He (juotes an actual experiment in coin tossing by
BufVoii. Out of 2,{)4S trials 1,{](»1 gave head at the first toss,
494 at the second, 2:>2 at the third, KJT at the fourth, oli at the
fifth, 29 at the sixth, 25 at the seventh, 8 at the eighth and at
the ninth. Computing the various mathematical expectations,
we find that the maximum value is found in the 25 sets with head
in the seventh toss, which gives a gain of 25 X r)4 = 1,()()(). The
most rare occiu*rence, the G sets of head in the ninth tb.row gives
a gain of X 25() = 1,5;)G, which is the next highest j^ain in all
the nine sets. De Morgan furthermore contends that if Bufi'on
had tried a thousand times as many games, the results would
not only have given more, but more yer game, arguing "that a
larger net would have caught not only more fish but more varieties
of fish; and in two millions of sets, we might have seen cases in
which head did not appear till the twentieth throw." Further-
more, "the player might continue until he had realized not only
any given sum, but any given sum per game." Therefore
according to De ]Morgan the mathematical expectation of a
player in a single game must be infinite.
CHAPTER VI.
PROBABILITY A POSTERIORI
39. Bayes's Rule. A Posteriori Probabilities. — The problems
hitherto considered liave all luid certain ])()ints in common.
Before entering upon the calculations of the mathematical
probability of the happening of the event in question, we knew
beforehand a certain complex of causes which operated in the
general domain of action. ^Ye also were able to separate this
general complex of productive causes into two distinctive minor
domains of complexes, of which one would bring forth the event,
E, while the other domain would act towards the production of
the opposite event, E. Furthermore we also were able to
measure the respective quantitative magnitudes of the two
domains, and then, by a simple algebraic operation, determine
the probability as a proper fraction. The addition and multi-
plication theorems did not introduce any new principles, but
only gave us a set of systematic rules which facilitated and
shortened the calculations of the relations between the different
absolute probabilities. The above method of determination
of a mathematical probability is known as an a j^riori determina-
tion, and such probabilities are termed a priori probabilities.
The problems treated in the preceding chapters have, nearly
all, been related to different games of chance, or purely abstract
mathematical quantities. The inorganic nature of this kind of
])roblems lias made it possible for us to treat them in a relatively
simple manner. Tn many of tlic problems, which we shall con-
sider hereafter, organic elements enter as a dominant factor and
make the analysis much more complicated and difhcult.
All social and biological investigations, which are of a much
larger benefit and practical vahic tlian the ])robUMns in games of
chance, lead often to a completely different categon' of ])robabil-
ity problems, which are known as " a posteriori ])robabilities."
In ])roblems where organic life enters into the cak'iilations, the
complex of productive causes is so varied and manifold, that
54
40] DISCOVERY AND HISTORY OF THE RULE. 55
our minds are not able to pifjeonhole the different productive
causes, j)lacing them in their proper domains of action. But we
know that such causes do exist and are the origin of the event.
If now, by a series of observations, we have noticed the actual
occurrence of the event, E (or the occurrence of the opposite
event E), the problem of the determination of an a posteriori
probability to find the probability that the event E originated
from a certain complex, say F. We must then, first of all,
form a complete hypothetical judgment of the form: E either
happens from the complexes i^i, or Fo, or F-i, ■ ■ • or F„. But we
must not forget that, in general, the different complexes F^
(a = 1, 2, •••, n) of the disjunction are not known a ])riori.
We must, therefore, determine the respective probabilities for
the actual existence of such disjunctive complexes F^. These
probabilities of existence for the complexes of causes are in
general different for each member, a fact which often has been
overlooked by many investigators and writers on a posteriori
probabilities, and which has given rise to meaningless and
paradoxical results.
40. Discovery and History of the Rule. — The first discoverer
of the rule for the computation of a posteriori probabilities by
a purely deductive process was the English clergyman, T. Bayes.
Bayes's treatise was first published after the death of the author
by his friend, Dr. Price, in Philosophical Transactions for 1763.
The treatise by the English clergyman was, for a long time,
almost forgotten, even by the author's own countrymen; and
later English writers have lost sight of the true " Bayes's Rule "
and substituted a false, or to be more accurate, a special case of
the exact rule, in the different algebraic texts, under the discus-
sion of the so called " inverse proba})iIities," a name which is due
probably to de Morgan, and which in itself is a great misnomer.
This point, presently, we shall discuss in detail.
The careless application of the exact rule has recently led to
a certain distrust of the whole theory of " a posteriori proba-
bilities." Scandinavian mathematicians were probably the first
to criticize the theory. In 1S79, Mr. J. Bing, a Danish actuary,
took a very critical attitude towards the mathematical principles
underlying Bayes's Rule, in a scholarly article in the mathe-
56 PROBABILITY A POSTERIORI. [41
matical journal Tidsskrift for Matematik. Bing's article caused
a sharp, and often heated, discussion among the older and younger
Danish mathematicians at that time; but his views seem to have
gained the upper hand, and even so great an authority on the
whole subject as the late Dr. T. Thiele, in liis well-known work,
" Theory of Observations "' (London, 19()o), refers to Bing's
article as "a crushing proof of the fallacies underlying the
determination of a posteriori probabilities by a purely deductive
method." As recently as 1908, the Danish writer on philosophy.
Dr. Kroman, has taken up cudgels in defense of Bayes in a
contribution in the Transadiuns of the Royal Danish Academy
of Science, which has done much towards the removal of many
obscure and erroneous vicMs of the older authors. Among
English writers, Professor Chrystal, in a lecture delivered before
the Actuarial Society of Edinburgh, has also given a sharp
criticism of the rule, although he does not go so dee])ly into the
real nature of the problem as either Bing or Kroman.
Despite Chrystal's advice to " bury the laws of inverse prob-
abilities decently out of sight, and not embalm them in text books
and examination papers " the old view still holds sway in recent
professional examination papers. It is therefore absolutely
necessary for the student preparing for professional examinations
to be acquainted with the theory. In the following ])aragraphs
we shall, therefore, give the mathematical theory of Bayes's
Rule with several examples illustrating its application to actual,
problems, together with a criticism of the rule.
41. Bayes's Rule (Case I). — {The different complexes of causes
producing the observed event, E, possess different a priori proba-
bilities of existence.) Let E denote a certain state or condition,
which can appear under only one of the mutually exclusive
complexes of causes: Fi, Fo, • • ■ and not otherwise. Let the
probability for the actual existence of F\ be ki and if F\ really
exists then let coi be the " productive probability " for bringing
forth the observed event, E {E being of a different natm-e from
F), which can only occur after the previous existence of one of
the mutually exclusive complexes, F. Let, in the same manner,
F2 have an " existence probability " of ko and a " productive
probability " of 0^2, F^ an existence probability of ks and a pro-
41 ] BA yes's rule. 57
ductive probability of 003 • • • etc. If now, by actual observa-
tion, we have noted that the event E has occurred exactly m
times in n trials, then the probability that the complex Fi was
the origin of E is:
Similarly that complex /^ ^vas the origin:
,, , , wY 1 , , ^ n — m
r^, =
and so on for the other complexes.
Proof. — Let the number of equally possible cases in the general
domain of action, which leads to one of tlie complexes F^, be /.
Furthermore, of these t cases let /i be favorable for the existence
of complex /'\,/2 for F2,/3 for F3, • • • , etc. Then the probabilities
for the existence of the different complexes F^ia = 1, 2, 3, • • • 71)
are:
/l f-2 /s ,. ,
Ki = y , K'2 = 'T , ^3 = Y ■ ' ■ respectively.
Of the/i favorable cases for complex Fi, Xi are also favorable for
the occurrence of E.
Of the/2 favorable cases for complex Fo, Xo are also favorable for
the occurrence of E.
Of the/3 favorable cases for complex F3, X3 are also favorable for
the occurrence of E.
The probability of the happening of E under the assumption that
Fi exists, i. e., the relative probability: PyJ^E), is:
COi
Al
"/l
^\
"A
or in general:
(a= 1, 2, 3, .-.).
The total number of equally likely cases for the simultaneous
occurrence of the event E with either one of the favorable cases
58 PROBABILITY A POSTERIORI. [41
for Fi, F2, Fz, ' ■ ■ is:
Xi + X2 + X3 + • • • = ^K.
The number of favorable cases for the simultaneous occurrence
of Fi and E is Xi, for the simultaneous occurrence of Fo and E,
X-:, • • • , etc. Hence: we have as measures for their corresponding
probabilities
But
Xi = oji • J\, Xo = 002 • fi,
, etc.,
and
/i = ^■l • i, f-i = K-z • t, ■
' y
etc.
Hence
Xi = COi • K\ • t, X> = CO-) • K-y
• t,
• • • , etc
Substituting these values in the above expression for Qi, Qo
we get:
as the respective probabilities that the obserxed event originated
from the complexes 7'\, Fo, F^, ■ • • . Such probabilities are called
a posteriori probabilities.
Let us now for a moment investigate the above expression for
Q\, Qi, • • • • The numerator in the expression for ^i is aci • coi.
But Ki is simply the a priori probability for the existence of F\
while coi is the a ])riori productive probability of bringing forth
the event observed from complex F\. The product /ci • coi is
simply the relative probability Pf^(E), or the probability that
the event E originated from Fi. In the denominator we have
the expression '^k^o)^ (a = 1,2, • • • n) which is the total proba-
bility to get A' from any of the com])lexes F^. From example 17
(Chapter IV) we know that the probability to get E exactly m
times from Fi in w total trials is:
7)1 = (2)k-i • c^rd - co,y-^
and the probability to get E from any one of the complexes, F,
43] BA yes's rule. 59
m times out of n is:
2/Ja = ( 2 ) l^K-a • a;/'(l - coj"— (a = 1, 2, 3, • • •)•
If, by actual observation, we know the event E to have happened
exactly m times out of 7i, then tlie a posteriori i)r()habiHty that
Fi was the origin is:
^^1= --r^TT (« = 1, 2, 3, . • .). (I)
^:)
The factorials I j in numerator and denominator cancel each
other of course. It will be noticed that, in the above proof, it
is not assumed that the a posteriori proba})ility is j)roportional
to the a priori probal)ility, an assumption usually made in the
ordinary texts on algebra.
42. Bayes's Rule (Case II). — {S,i)eclnl Case. The a priori
probabilities of existence of the different complexes are equal.)
Sometimes the different complexes F may be of such special
characters tha!; their a priori probabilities of existence are equal,
i. e.,
Ki = K-2 = K3 = K:4 • • • Kn-
In this case the equation (I) simjily reduces to:
cor(i — (oi)""^
Equation (I) gives, however, the most general expression for
Bayes's Rule which may be stated as follows:
// a definite observed event, E, can originate from a certain series
of mutually exclusive complcres, F, and if the actual occurrence of
the event has been observed, then the probability that it originated
from a specified complex or a specified group of complexes is also
the " a posteriori " probability or probability of existence of the
specified complex or group of complexes.
43. Determination of the Probabilities of Future Events
Based Upon Actual Observations. — It happens frequently that
60 PROBABILITY A POSTERIORI. [43
our knowledge of the general domain of action is so incomplete,
that we are not able to determine, a priori, the ])robability of the
occurrence of a certain expected event. As we already have
stated in the introduction to a posteriori probabilities, this is
nearly always the case with problems wherein organic life enters
as a determining factor or momentum. But the same state of
affairs may also occur in the category of i)r()l)lcms relating to games
of chance, which we have hitherto considered. Sui)pose we had
an urn which was known to contain white and black balls only,
but the actual ratio in which the balls of the two different colors
were mixed, was unkiu)wn. With this knowledge beforehand,
we should not be able to determine the ])robability for the drawing
of a white ball. If, on the other hand, we knew, from actual
experience by repeated obser\'ations, the results of former draw-
ings from the same urn when the conditions in the general domain
of action remained unchanged during each separate drawing, then
these results might be used in the determination of the prob-
ability of a specified event by future draA\ings.
Our problem may be stated in its most general form as follows:
Let Fa denote a certain state or condition in the general domain
of action, which state or condition can appear onj}' in one or the
other of the mutually exclusive forms: Fi, F-y, /"s, • • •. and not
otherwise. Let the probability of existence of Fi, F-2, F^, • • • be
K\, K2, K'3, • • • respectively, and when one of the complexes F\, Fo,
Fz, • • • exists (occurs) let coi, coo, C03, • • • be the respective pro-
ductive probabilities of bringing forth a specified event, E.
If now, by actual observation, we know the event, E, to have
happened exactly ni times out of n total trials (the conditions in
the general domain of action being the same at each individual
trial), what is then the probability that the event, E, will happen
in the (n + l)th trial also?
By Bayes's Rule we determined the "a posteriori " probabili-
ties or the probabilities of existence of the complexes Fi, Fi, • • ■
as:
fl ~ V.r . /., m/l , . \n—m ' ^2 ~
{a= 1, 2, 3, •••).
In the in -f l)th trial E may happen from any one of the mutually
44] APPLICATION OK BAVEs's RULE. 61
exclusive complexes: /'\, F-i, P\, ■ ■ ■ whose respective probabilities
in producing the event, E, are coi, ooo, (^3, •••• The addition
theorem then gives us as the total ])robabiIity of the occurrence
of Em the (/? + l)th trial:
R^ = ^P,M^) = (?i . coi + (?, . CO, + Q, • C03
SK' „-a,„"'(l-coJ"--a). ^ ^ ^ ^ ^ (HI)
^K^ -CO, U — CO J
If the a priori probabilities of existence are of equal magnitude
(Case II) the factors k in the above expression cancel each other
in numerator and denominator and we have
^ ScO.-d - CO J "-"CO.
2co/'(l -coj— • • ^^'^
44. Examples on the Application of Bayes's Rule. — Example
21. — An urn contains two balls, white or l)lack or both kinds.
What is the proba})ility of getting a white ball in the first draw-
ing, and if this event has happened and the ball replaced, what
is then the probability to get white in the following drawing?
Three conditions are here possible in the urn. There may be 0,
1, or 2 white })alls. Each hyi)othelical condition has a proba-
bility of existence equal to \, and the producti^'e probabilities
for white are 0, | and 1 respectively. The total probability to
get white is therefore:
Pi = I • + i • i + i • 1 = i
If we now draw a white ball then the probabilities that it came
from the complexes: Fi, ¥■>, F3, respectively, are:
Q . 1 1^1 1^1
2' 6 • 2' 3
These are also new existence probabilities of the three proba-
bilities. The probability for white in second drawing is therefore .
(0-^^)0+(J-^|)§+(i^i)l = f.
This solution of the problem is, however, not a unique solution,
because it is an arbitrary solution. It is arbitrary in this respect,
that we have without further consideration given all three com-
plexes the same probability of existence, \. We shall discuss
62 PROBABILITY A POSTERIORI. [45
this part of the question under the chapter on the criticism of
Bayes's Rule.
Example 22. — iVn urn contains fi^'e balls of which a part is
known to be white and the rest black. A ball is drawn four
times in succession and replaced after each drawing. By three
of sucli (h-awiiiii's a white ball was obtained and l)y one drawing
a black l)all. What is the probability that we will get a white
ball in the fifth drawing?
In regard to the contents of the urn the following four hypoth-
eses are possible:
¥\: 4 Avhite, 1 black balls,
¥.: 3 " 2 "
l\\ 2 " 3 " "
¥,: 1 " 4 "
Since we do not know anything about the ratio of distribution
of the different colored balls, we may by a direct application of
the principle of insufficient reason regard the four complexes as
equally probable, or:
K\ = Ko = K3 = K'4 = J.
If either Fi, Fo, F3 or Fi exists, the respective productive
probabilities are:
wi = I, C02 = f, aj3 = f , C04 = i.
By a direct substitution in the formula:
R =
2co/'(l - ojj"-^
(a = 1,2, 3, 4) for n = 4 and m = 3 we get:
p _ (miw + gram + amm + anm) ^ ,,
^tm) + (im) + (im) + (im) ^*"
45. Criticism of Bayes's Rule, — In most English treatises on
the theory of cliaiice the " a posteriori " determination of a
mathematical probability is discussed under the socalled " in-
verse probabilities." This somewhat misleading name was prob-
ably first introduced by the eminent PvUglish mathematician and
actuary, Augustus de Morgan. In the opening of the discussion
45] CRITICISM OP" BAYEb'.S RULE. 63
of a posteriori ])r()l);il)ilitic> in the tiiird chai)tcr of his treatise,
" An Essay on rrol);il)ilities/" de Morgan says: " In the preceding
chapter, we liave calcidated the chances of an event, knowing the
circumstances under which it is to happen or faih We are noAV
to place ourselves in an inverted position, we know the event,
and ask what is the probabiHty which results from the event
in favor of any set of circumstances under which the same might
have happened." Is this now an inverse process? By the a
priori or — as de Morgan prefers to call them, — the direct prob-
abilities, we started from a definitely known condition and de-
termined the probability for a future event, E, or what is the same,
the probability of a specified future state of affairs. Here we
start knowing the present condition and try to determine a past
condition. The ])rocess apparently a])pears to be the inverse of
the former, although they both are the same. We possess a
definite knowledge of a certain condition and try to determine
the probability of the existence of a specified state of affairs, in
general different from the first condition, but whether this state
of affairs occurred in the past or is to occur in the future has no
bearing on our problem. In other words, time does not enter
as a determining factor. And even if we were willing to admit
the two processes of the determination of the different probabil-
ities to be inverse, the probabilities themselves can not be said
to be inverse. Nevertheless, this misleading name appears over
and over again in examination papers in England and in America
as a thoroughly embalmed corpse which ought to have been
buried long ago. What is really needed, is a change of customary
nomenclature in the whole theory of probability. Instead of
direct and inverse, a priori and a posteriori probabilities, it would
be more proper to si)eak about " prospective " and " retro-
spective " probabilities in the application of Bayes's Rule. All
probabilities are in reality determined by an empirical process.
That there is a certain probability to throw a six with a die we
only know after we have formed a definite conception of a die.
The only probabilities which we perhaps rightly may name a
priori are the arbitrary probabilities in purely mathematical
problems where we assume an ideal state of affairs. " There
is," to quote the Danish writer on logic. Dr. Kroman, " really
64 PROBABILITY A POSTERIORI. [46
more reason to doubt the a priori than the a posteriori probabil-
ities, and it would be more natural and also more exact in the
application of Bayes's Rule to speak about the actual or original
and the new or gained probability."
The discussion above has really no direct bearing on Bayes's
Rule but was introduced in order to give the student a clearer
understanding of the main principles underlying the whole deter-
mination of a posteriori probabilities by means of actual experi-
mental observations, and also to remove some obscure points.
From his ordinary mathematical training every student of mathe-
matics has an almost intuitive imderstanding of an inverse process.
Naturally when he encounters again and again the customary
heading: " inverse probabilities " in text-books he obtains from
the very start — almost before he starts to read this particular
chapter — an inverse idea of the sul)ject instead of the idea he really
ought to have. Nowhere in continental texts on the theory of
probabilities, will the reader be able to find the words direct and
inverse applied in the same sense as in English texts since the
introduction of these terms by de Morgan. We shall advise
readers who have become accustomed to the old terms to pay
no serious attention to them.
46. Theory Versus Practice. — In § 41 we reduced Bayes's
Rule to its most general form:
^ - -K. • cord - coj"- ^" - i. -, -^, • • •)•
This is an exact expression for the rule, but it is at the same
time almost impossible to employ it in ])ractice. Only in a few
exceptional cases do we know, a priori, the different values of the
often numerous probabilities of existence k„, of the complexes
F^, and in order to api)ly the rule with exact results we require
here sufficient facts and information about the dilfcrcnt com-
plexes of causes from which the observed event. A', originated.
Bayes deduced the rule from special examples resulting from
drawings of balls of different colors from an urn where the different
complexes of causes were materially existent. The i)robability
of a cause or a certain com])Iex of causes did not here mean the
probability of existence of such a complex but the i)robability
■46] THEORY VERSUS PRACTICE. 65
that the observed event originated from this particuhir complex.
In order to eiucichite this statement we give following sinipU;
exami)le:
Example 23. — A bag contains i coins, of which one is coined
with two heads, the other three having both head and tail. A
coin is drawn at random and tossed four times in succession and
each time head turns up. What is the probability that it was
the coin with two heads?
The two complexes /\ and Fo, whicii may produce the event,
E, are: Fi, the coin with two heads, and /-'•>, an ordinary coin.
The probability of existence of /\ is the probability of drawing
the single coin with two heads which is equal to j, the probability
of existence for the other complex, F-y, is equal to |. The
respective productive probabilities are 1 and ^. Thus ki = ^,
Ko = 4 , coi = 1 and coo = ^. Substituting these values in formula
(I) (n = 4, m = 4), we get:
Q = axv)^{\xv + ix ay) = i-^u = H-
But in most cases we do not know anything about the material
existence of the complexes of causes from which the event, E,
originated. On the contrary, we are forced to form a hypothesis
about their actual existence. To start with a simple case we take
example 21 of § 44.
We assumed here three equally possible conditions in the urn
before the drawings, namely the presence of 0, 1, or 2 white balls.
From this assumption we foiuid the probability to get a white
ball in the second drawing, after we had i)re\'iously drawn a white
ball and then put it back in the urn before the second drawing,
to be equal to |. As we already remarked, this solution is not
unique because it is an arbitrary solution. It is arbitrary to
assign, without any consideration whatsoever, ^ as the probability
of existence to each of the three conditions. Let us suppose
that each of the two })alls bore the numbers 1 and 2 respectively.
W^e may then form the following equally Hkely conditions:
?>i6o, hiir-2, h-iU'i, iViWo,
each condition having an a })riori probability of existence equal
to J and a productive probability for the drawing of a white
6
66 PROBABILITY A POSTERIORI. [46
ball equal to: 0, ^, ^ and 1 respectively. Thus:
K'l = K2 = 'v'S — '<4 = 4
and
The respective a posteriori probabilities, that is the new or
gained probabilities of the four hyi)othetical conditions, become
now by tiie ai)plieati()n of Bayes's Rule (Formula II):
Qi= I f?2 = I - 2, r^3 = ^ - 2, (?4 = i
Hence the pr()l)abiUty for white in the second drawing is:
/ -co "'(1 - a)J"~'"co„\
(Formula IV: 7? = V^lvi ^\;r^)
\ 2a)„'"(l - wj"-'" /
2a),-(l - wj'
7? = - 2 + (i - 2) + (i - 2) + (1^2)
In the first solution we got I for the same probability. Which
answer is now the true one? Neither one I The true answer to
the problem is tluit it is not given in such a form tliat the last
question — the probability of getting a wliite ])all in the second
drawing — may be settled without any doubt. The answer must
be conditional. Following the first hypothesis we got |, while
the second hypothesis gives | as the answer.
We next proceed to example 22 which is abnost identical in
form to the first one, the only difference l)cing a greater variety
of hypothetical conditions. We started here with the folh)wing
four hypotheses:
Fi: 4 white, 1 black ball, lu: 3 wliite, 2 black, F.,: 2 white, 3
black and F4: 1 white and 4 black balls, assigning | as the hy-
pothetical existence ])roba})ility.
By marking the 5 balls similarly as in the last example, with
the numbers from 1 to 5 we may form the comj)lexes:
Fi: 4 white and 1 black ball in (1) ways,
Fo: ?> " " 2 " balls" il) "
F3: 2 " " .3 " " " (|) "
^4=1 " " 4 " " " (f) "
This gives us a total of 5 + 10 + 10 + 5 = 30 different
complexes. Assuming all of these complexes equally likely
47] PROBABILITIES FXPRESSED BY INTEGRALS. 67
to occur, we got following probabilities of existence and pro-
ductive probabilities:
Ki = K2 = K3 = Ki = ' ' ' — >^so — '5^
coi = CO) = C03 = o).} = ojs = I (Productive prob. for Fi)
CO6 = C07 = coh = • • • = Wis = I (Productive prob. for F2)
CO 16 = 00 17 = • • . = 0)25 = I (Productive prob. for Fs)
o)->& = CO27 = C028 = CO29 = C1J30 = ^ (Productive prob. for F4).
The total probability of getting a white ball in the second
drawing is now —^t,\ ^ — (a = 1, 2, 3, • • •, 30).
Sco^ll-ooJ
Actual substitution of the above values of co in this formula
gives us the final result as: R = ||.
47. Probabilities Expressed by Integrals. — By making an ex-
tended use of the infinitesimal calculus Mr. Bing and Dr. Kroman
in their memoirs arrived at much more ambiguous results through
an application of the rule of Bayes. Starting with the funda-
mental rule as given in equation (I) in § 41, we may at times en-
counter somewhat simpler conditions inside the domain of
causes. The total comi)lex of actions may embrace a large
number of smaller sub-comi)lexes construed in such a way that
the change from one complex to another may be regarded as a
continuous process, so that the productive probabilities are
increased by an infinitely small quantity from a certain lower
limit, a, to an upper limit, b. Denoting such continuously in-
creasing probabilities by i' and the corresponding small proba-
bilities of existence by vdv, we have as the total probability of
obtaining E from any one of the minor complexes with a pro-
ductive probability between a and ^ (a ^ a, ^ ^ b)
p = I uvdv.
•'a
The probability that when E has happened it originated from
one of those minor complexes, or the probability of existence of
some one of those complexes is:
I uvdv
p _ ^/a
uvdv
i:
68
PROBABILITY A POSTERIORI.
47
The situation may be still more simplified by the following con-
siderations. In the continuous total complex between the limits
a and h we have altogether situated {h — a)ldv individual minor
complexes. If we assume all of these complexes to possess the
same probability of existence, we must have:
iidv
dv
b — a'
The two formulas then take on the form:
I
vdv
and
A still more specialized form is obtained by letting a = and
b = I which gives:
-f
vdv and P —
.1
7"
vdv
vdv
The above formulas may perhaps be made more intelligible
to the reader by a geometrical illustration.
Let the various productive ])robabirities, i\ be i)lotte(l along the
A" axis in a Cartesian coordinate system in the interval from a
to b (a < h). To any one of these probabilities say ^v there
corresponds a certain probability of existence, iir, represented
by a }' ordinate. In the same manner the next following pro-
ductive probability, jv+i, will have a probability of existence
47] I'liOHABILITIES EXPRESSED BY INTEGRALS. 69
represented })y an ordinate ?/r+i. It is now possible to represent
the varions //'s by means of areas instead of line ordinates. Thus
the probability of existence, Ur, is in the figure represented by
the small shaded rectangle, with a base equal to
and an altitude of Ur, the total area being equal to AvrUr. That
this is so, follows from the well-known elementary theorem from
geometry that areas of rectangles with equal bases are directly
proportional to their altitudes. The sum of the difi'erent u'h is
thus in the figure re])resented as the sum areas of the various
small rectangles in the staircase shajx-d histograph. Now ac-
cording to our assum])tion v is a continuous function in the interval
from a to b. We may, therefore, divide this interval, b — a,
into n smaller equal intervals. Let
b — a
Vr+l — l\ = ACr =
n
be one of these smaller divisions. By choosing n sufficiently
large, (b — a);n or Av becomes a very small quantity and by
letting w approach infinity as a limiting value we have
,. /^ — « ,.
inn u — — = hm uAv = udv.
In this case the histograph is replaced by a continuous curve and
udv is the probability of existence that the i)roductive probability
is enclosed between /" and i- + dc.^
The probability to get E from any one of the complexes is
evidently given by the total area of the small rectangles, or in
the continuous case by means of the integral:
r
uvdv
1 A more rigorous analysis would be as follows: We plot along the abscissa
axis intervals of the length t so that the middle of the interval has a distance
from the origin equal to an integral nmltiple of e. If now e is chosen suffi-
ciently small, we may regard the i)robability of existence of ;/, for values of
the variable v between rt — \t and re + 5^ as a constant and the probability
that V falls between the limits re — \e and r« + |e may hence be expressed as
eur. When e approaches as a limiting value this expression becomes vdv.
See the similar discussion under frequency curves.
70 PROBABILITY A POSTERIORI. [48
In the same way the probabiHty that E originated from any
of the complexes between a and /3 is:
r
IV dv
f.
b
uvdv
The special case a = and 6=1 needs no further commentary.
We are now in a position to consider the examples of Bing and
Kroman, Any student familiar with multiple integration will
find no difficulty in the following analysis. For the benefit of
readers to whom the evaluation of the various integrals may seem
somewhat difficult, we may refer to the addenda at the close of
this treatise or to any standard treatise on the calculus as, for
instance, Williamson's " Integral Calculus."
48. Example 24. — An urn contains a very large number of
similarly shaped balls. In 10 successive drawings (with replace-
ments) we have obtained 7 with the number 1, 2 with the number
2, and one having the number 3. Wliat is the probability to
obtain a ball with another number in the following drawing?
We must here distinguish between 4 kinds of l)alls, namely
balls marked 1, 2, 3, or " other balls." A general scheme of
distribution of the balls in the urn may be given through the
following scheme:
nx balls marked with the number 1,
ny
o
nz " " " " " 3 and
nt = n{\ — X — y — z) other balls.
Here .r, y, z and / represent the respective productive probabil-
ities. If we now let all such probabilities assume all possible
values between and 1 with intervals of 1/??, we obtain the pos-
sible conditions in the total complex of actions. Each of these
conditions has a probability of existence, .v, and the ])roductive
probabilities x, y, z, and 1 — .r — // — z. The original probability
for 7 ones, 2 twos and 1 three in 10 drawings is:
10'
48] EXAMPLE 24. 71
Now when n is a very l;ir<fe iiuinher the interval ]ln becomes a
very small quantity, and we may approximately write:
s = udxdydz,
and also write the alcove sum as a triple integral:
10! f C'' n
^ ^ 7i->ii ' u ■ x'' ■ y~ - z ■ dx ■ dy • dz,
where
p = 1 — .1- and q — \ — x — y.
If now the above event has happened, then the probal)ility to get
a different marked ball in the 11th drawing is:
ml
u ' .1-^ • y- • 2(1 — X — y — z) ■ dx • dy • dz
H =
I I I '' ■ •^■" • i/' ' 2 • dx ■ dy ■ dz
It is, however, quite impossible to evaluate the above integral
without knowing the form of the function ?/; but unfortunately
our information at hand tells us absolutely nothing in regard to
this. Perhaps the balls bear the numbers 1, 2 and 3 only, or
perhaps there is an equal distribution up to 10, ()()() or any other
number. Our information is reall\ so insufficient that it is quite
hopeless to atteni])t a calculation of the a posteriori probability.
IMany adherents of the inverse probability method venture,
however, boldly forth with tlie following solution based upon the
perfectly arbitrary hypothesis that all the ?/'s are of equal magni-
tude. This gives the special integral:
m.r" • y- • 2(1 — X — y — z), dx - dy • dz
— -
v'O tVo «^0
y~ ' z ■ dx • dy • dz
where once more it must be remembered that
x+ y+z^l.
In this case the limits of x are and 1, those of y are and I — x
and those of 2 are and 1 — .r — y.
This is a well-known form of the triple integral which may be
evaluated bv means of Dirichlet's Theorem:
72 PROBABILITY A POSTERIORI. [49
^1 ^i-x ^.-.-j, r(6)r(//or(»)
Jo Jo Jo r(H-6+m+^0
(See Williamson's Calculus.)
Remembering the well-known relation between gamma func-
tions and factorials, viz. T(n + 1) = ??!, we find by a mere
substitution in the integral, the value of the probability in
question to be 1:14. Another and cquidly plausible result is
obtained by a slightly different wording of the problem.
Ten successive drawings have resulted in balls marked 1, 2,
or 3. What is the probability to obtain a ball not bearing such
a number in the 11th drawing? This probability is given by
the formula.
r v'%1 - v)dv
^ = 1 • 1*^
Jo
Quite a different result from the one given above.
49. Example 25 — Bing's Paradox. — A still more astonishing
paradox is produced by Bing when he gi^•es an example of Bayes's
Rule to a problem from mortality statistics. A mortality table
gives the ratio of the number of persons living during a certain
period, to the number living at the beginning of this period,
all persons being of the same age. By recording the deaths
during the specified period (say one year) it has been ascertained
that of .v persons, say forty years of age at the beginning of the
period, m have died during the period. The observed ratio is
then (.9 — m)ls. If s is a very large number this ratio may (as
we shall have occasion to prove at a later stage) be taken as an
approximation of the true ratio of probability of sur\-ival during
the period. If .v is not sufficiently large the belie\'(M's in the inverse
theory ought to be able to evaluate this ratio by an ap])lication
of Bayes's Rule, by means of an analysis similar to the one as
follows:
Let // be the general symbol for the probability of a forty-
year-old ])(Tson being alive one year from hence. Eadi of such
persons will in general be subject to different conditions, and the
general symbol, y, will therefore have to be understood as the
49] EXAMPLE 25. bing's paradox. 73
symbol for all the ])()ssil)l(' productive probability values changing
from to 1 by a continuous process.
Assuming s a very large number each condition will have a
probability of existence equal to udy. We may now ask: What
is the probability that the rate of survival of a group of s persons
aged 40 is situated between tiie limits a and /3?
The answer according to Baves's Rule is:
I
?/""'(! — y)"'ndy
.' ^ . (I)
7/"^(l — y)"'udy
Let us furthermore divide the whole year into two equal parts
and let y\ be the probability of surviving the first half year,
2/2 the probability of surviving the second half, and 7/1 • dy\,
111 • dy2 the corresponding probabilities of existence. Then the
respective a posteriori probabilities for ?/] and ?/2 are:
2/i«-'"'(l - yiT' uidyi
I
1
?/i^~™>(l — yi)"''uidyi
and
j!/2^~"'(l — y'iY''ii'idy2
f
(nil + m-i — m)
?/2*~™ (1 — y2)"'-vidy2
'0
(nil and m2 represent the number of deaths in the respective half
years.) The probability that both ?/i and y^ are true is then
according to the multiplication theorem:
^/i-'-^Hl - yi)""n\dyiy-f~'"{l — y-d'^-U'^dyi
j y^^-m^ii - y{)-^Hiidyi ( 1/2*^(1 - y2Thi2dy2
Jo Jo
where y = y\ • y-i.
The probability that the probability of survival for a full
year, y, is situated between the limits a and jS is therefore:
a
"(1 - ?/i)""//2'~"'(l - 2/2)"''Mi • ?/2 • dyx ■ dlj2
I .vi^^'(l - yi)""iti(Jy\ I y-i^'d — yiT-n^dy
Jq ' ' Jo
(11)
74 PROBABILITY A POSTERIORI. [49
where the Hmits in the double integral in the numerator are de-
termined by the relation:
Choosing the principle of insufficient reason as the basis of
our calculations, merely assuming that all possible events are, in
the absence of any grounds for inference, equally likely, the
various quantities cx])ressed by the general symbol, ?/, become
equal and constant and cancel each other in numerator and
denominator, which brings the a posteriori probabilities ex-
pressed by (I) and (II) to the forms:
£
y^il - ijTdy
J
(III)
and
//
yr'^'H - y,ryr'"'-'"^-il - y2)'^dy, ■ dy^
f ^I'-'-Ki - yiY"' \ y-f-^0^ - y2)'""-dy,-dy^
Jq *Jo
(IV)
where the limits in the numerator in the latter expression are
determined by the relation : a < y\y2 < |8.
Letting
y
yi = ~~
and then
1 — 2/1 = 2(1 - y)
this latter expression may after a simple substitution be brought
to the form:
fyr-'Hl - yirdyrj y.'-^il - y.rdy^
(V)
(See appendix.)
]\Ir. Bing now puts the further question : A\Tiat is the probability
that a new person forty years of age, entering the original large
49] EXAMPLE 25. bing's paradox. 75
group of s persons, will survive one year, when we assume
mi = ni'i = 0? (Ill) gives the answer:
X
Formula (V), on the other hand, gives us
I in" (hi I I U2(hj2
^= (::;)•
As the above analysis is perfeetly general, we might equally
well have ap])lied it to each of the semi-annual periods, which
would give us an a posteriori probability of survival equal to
I I ., ) for each half year, or a compound probability of
I I .^ I for the whole year. Extending this process it is
easily seen that by dividing the year into parts, we shall have
(5 4- 1 \ "
-3^-^ I as the final probability a posteriori that a forty-year-
old person will reach the age of forty-one. By letting n increase
indefinitely the above quantity a])])roac]ies as its limiting
value and we obtain thus the ])arad()x of Bing:
If, among a large group of s cqudlh/ old prrsoih'i, we have observed
no deaths during a full calendar year then another person of the
same age outside the group is sure to die inside the calendar year.
This is evidently a very strange result, and yet, working on
the basis of the principle of insufficient reason, the mathematical
deductions and formula exhibit no errors.
Mr. Bing disposes of the whole matter by simply denying the
validity and existence of a ])()steriori probabilities. Dr. Kroman
on the other hand defends Bayes's Rule. " Mathematics,"
Kroman says, " is — as Huxley has justly remarked — an ex-
ceedingly fine mill stone, but one must not expect to get wheat
flour after having put oats in the quern." According to the
7G I'ROHABILITY A POSTERIORI. [50
Danish scliolar the paradox is due to tlie use of a wrong formula.
We ought to have used the general formula (11) instead of formula
(V) whieh is a special case. In the general formula we encounter
the functions ?/, denoting the probability existence of the various
productix^e probabilities y. As we do not know anything about
this function // it is lu)peless to attempt a calculation. This
brings the criticism down to the fundamental question whether
we shall build the theory of probabilities on the principle of
" cogent reason " or the principle of " insufficient reason."
50. Conclusion. — Contradictory results of a similar kind to
the ones given above have led several eminent mathematicians
to a complete denunciation of the laws underlying a posteriori
probabilities. Professor Chrystal, especially, becomes extremely
severe in his criticism in the previously mentioned address before
the Actuarial Society of Edinburgh. He advises " practical
people like the actuaries, much though the}' may justly respect
Laplace, not to air his weaknesses in their annual examinations.
The indiscretions of a great man should be quietly allowed to be
forgotten." Although one may heartily agree with Professor
Chrystal's candid attack on the belief in authority, too often
prevailing among mathematical students, I think — aside from
the fact that the rule was originally given by Bayes — that the
great F'rench savant has been accused unjustly as the following
remarks perhaps may tend to show.
In our statement of Bayes's Rule, we followed an exact mathe-
matical method, and the final formula (I) is theoretically as
correct as any previously demonstrated in this work. The
customary definition of a mathematical probability as the
ratio of equally favorable to coordinated possible cases, is not
done away with in this new kind of probabilities; the former are
found in the numerator and the latter in the denominator; and
if we take care that each of the particular formulas, with its
definite requirements, is applied to its particular case, we do not
go beyond ])ure mathematics or logic. But are we able to get
complete and exact information about these re({uircnients? In
the example of the tossing of a coin with two heads, this informa-
tion was at hand. Here we were able to enumerate exactly the
difi'erent mutually exclusive causes from which the observed
50] CONCLUSION. 77
event originated. We were also able to determine the exact
quantitative measures for the probabilities, k, that these com-
plexes existed as well as the different productive pr()l)id)ilities, a;.
Here the most rigid requirements could be satisfied, and the rule
gave therefore a true answer.
In the other examples we encountered a different state of
affairs. Here we were not able to enumerate directly the dif-
ferent complexes of causes from which the event originated, but
were forced to form different and arbitrary hyi)()tlieses about the
complexes of origin, /', and each hypothesis gave, in general, a
different result. Furthermore, we assumed a priori that the
different probabilities of the actual existence of the complexes
were all equal in magnitude, and it was, therefore, the special
formula (H) we emi)l()yed in the determination of the a posteriori
probabilities. In this formula, the different ks do not enter at
all as a determining factor; only the i)roductive probabilities, co,
are considered. The assumption that all the k's are equal in
magnitude is based upon the principle of insufficient reason, or
as Boole calls it, " the equal distribution of ignorance."
The principle of equal distribution of ignorance makes in the
case of continuously varying productive probabilities, v, the
function, //, of the proba])ilities of existence of the various
complexes equal to a constant quantity. In other words, the
curve in Fig. 1, is replaced by a straight line of the form, it — k.
Now, as a matter of fact, we possess in most cases, some partial
knowledge of the complexes of action producing the event in
question. This partial knowledge — although far from complete
enough to make a rigorous use of formula (I) — is nevertheless
sufficient to justify us in discarding completely any general
hypothesis assuming such simple conditions as above. Such
partial knowledge is, for instance, found in the Paradox of Bing.
Here the rather absurd hypothesis was made that the possible
values of the probability of surviving a certain period were
equally probable. In other words, it is equally probable that
there will die 0, 1, 2, • • -, or s persons in the particular period.
" Common sense, however, tells us that it is far more probable
that, for instance, 90 jicr cent, of a large number of forty-year-old
persons will survive the i)eriod than no one or every one will die
78 PROBABILITY A POSTERIORI. [50
in the same period " (Kromaii). The indiscreet use of formula
(II) therefore naturally leads to paradoxical results. On the
other hand, the fallacy of the luippy-j:;o-lucky computers, em-
ploying the special case (II) of Bayes's Rule, as well as the critics
of Laplace, lies in their failure to make a proper distinction
between " equal distribution of ignorance " and " partial cogent
reason," which latter expression properly may be termed " an
unequal distribution of ignorance." If, despite the actual
presence of such unequal distribution of ignorance, we still insist
in using the special formula (II), which is only to be used in the
case of an equal distribution of ignorance, it is no wonder we
encounter ambiguous answers. Not the rule itself, its discoverer,
or Laplace, but the indiscreet computer is the one to blame.
Messrs. Bing, Venn and Chrystal, in their various criticisms, have
filled the quern with some rather "wild oats" and expected to
get wheat flour; and that one of those critics in his disappoint-
ment in not getting the expected flour should blame Laplace, is
hardly just.
So much for the principle of " equal distribution of igno-
rance." It may be of interest to see how matters tm-n out when
we like von Kries insist upon the principle of " cogent reason "
as the true basis of our computations. The reader will quite
readily see that a rigorous application of the Rule of Bayes in its
most general form as given by fornuila (I) really tacitly assumes
this very principle. In formula (I), we require not alone an
exact enumeration of the various complexes from which the
observed event may originate, but also an exact and complete
information about the structure of such complexes in order to
evaluate their various probabilities of existence. If such informa-
tion is present, we can meet even the most stringent requirements
of the general formula, and we will get a correct answer. But
in the vast majority of cases, not to say all cases, such information
is not at hand, and any attempt to make a computation l)y means
of Bayes's Rule must be regarded as hopeless. We may, how-
ever, again remark that very seldom we are in complete ignorance
of the conditions of the complexes, w^hich is the same thing as
saying that we are not in a position to employ the principle of
equal distribution of ignorance in a rigorous manner. From
50] CONCLUSION. 79
other experiments on tlie same kind of event, or from other
sources, we may have attained some partial information, even if
insufficient to employ the ])rinciple of cogent reason. Is such
information now to be comi)letely ignored in an attempt to give
a reasonable, although approximate answer? It is but natural
that the mathematician should attempt to obtain as much of
such information as possible and use it in the evaluation of the
various probabilities of existence. Thus for instance, if, in the
Paradox of Bing, we had observed that the probability of survival
for a forty-year-old person never had been below .75 and never
above .95, it would be but reasonable to substitute those limits
in their proper integrals in order to attain an approximate answer.
To illustrate this somewhat subjective determination of an a
posteriori probability, we take another example from the memoirs
of Bing and Kronian.
Example (24)- — A merchant receives a cargo of 100,000 pieces
of fruit. If every single fruit is untainted, the value of the cargo
may be put at 10,000 Kroner. On the other hand, any part of
the cargo more or less tainted is considered worthless. The
merchant lias never before received a similar cargo and does not
know how the fruit has been afl'ected by travel. As samples, he
has selected 30 pieces picked at random from the cargo and all
samples proved to be fresh. lie asks a mathematician what
value he can put on the cargo.
If the mathematician uses the special formula (II), assum-
ing an equal distribution of ignorance, therefore assuming that
it is equally probable that for example none, 5,000 or all the
individual pieces of fruit were untainted, the answer is:
10,000 ^^ = 9687.5 Kroner.
( f^¥v
Jo
If we use the trne rule, the a posteriori probability of the whole-
someness of the cargo is given by the integral:
>i
i
1
80 PROBABILITY A POSTERIORI. [50
where v is tlie general expression for a possible probability of
wholesomeness between and 1 and udv the corresponding proba-
bility of existence. Now if the mathematician has no complete
information as to this particular function, ?/, it would be foolish
if him to attempt a calculation, since the hy])othcsis of an equal
probability of existence for all possible values of v evidently
gives an arbitrary and perhaps a very erroneous result. On
the other hand, the computer may possibly have access to some
partial information. Perhaps tlie merchant has rec(>ived fruit
of a similar kind or heard about cargoes of this particular kind
of fruit received by other dealers. If now the merchant were
able to inform the computer that in a great number of similar
cases the probability of wholesomeness had been between 0.9
and 1 with an approximately even distribution, while it never
had been below 0.9, then nothing would hinder the mathematician
to present the following comi)utation:
^''' . = 0.9726
v^^le
and tell the merchant that on the basis of the information given
9,726 Kroner would be a fair price for the cargo.
This is really the point of view taken by the English mathe-
matician. Professor Karl Pearson, one of the ablest writers on
mathematical statistics of the present time, when he says: "I
start, as most writers on mathematics liave done, with ' the
equal distribution of ignorance ' or I assume the truth of Bayes's
Theorem. I hold tliis theorem not as rigidly demonstrated, but
I think with Edgeworth that the hypothesis of the equal dis-
tribution of ignorance is, within the limits of ])ractical life,
justified l)y our experience of statistical ratios, which are unknown,
i. e., such ratios do not tend to cluster markedly round any
])articuhu- j)()iiit."
To sum up the above remarks: Theoretically Bayes's Rule is
true. If we are able to enumerate and determine the probabilities
of existence of the complexes of origin it will also give true
results in practice. If we are justified in assuming the principle
50 ] CONCLUSION. 81
of " insufficicMit reason " or " eciual distrihiition of i<^iioraiice "
as the basis for our calculations, formula (II) may be employed
with exact results after a rigid enumeration of the complexes.
If the principle of " cogent reason " is required as the basis, an
exact computation is in general hopeless, and we can only after
having obtained ])artial subjecti\'c infoimation give an approxi-
mate answer.
Witli these remarks we shall conclude the elementar\' dis-
cussion of the merely theoretical ])art of the subject, '^riie follow-
ing chapters require in most cases a knowledge of the infinitesimal
calculus, and many of the questions discussed above will appear
in a new and instructive light by this treatment.
CHAPTER VII.
THE LAW OF LARGE NUMBERS.
51. A Priori and Empirical Probabilities. — In the previous
chapters we limited ourselves to the discussion of such mathe-
matical probabilities, where we, a priori, on account of our
knowledge of the various domains or complexes of actions, were
able to enumerate the respective favorable and unfavorable
possibilities associated with the occurrence or non-occurrence of
the event in question. " The real importance of tHe theory of
probability in regard to mass phenomena consists, however,
in determining the mathematical relations of the various proba-
bilities not in a deductive, but in an empirical manner — without an
a priori exhaustive knowledge of the mutual relations and actions
between cause and effect — by nteans of statistical enumeration
of the frequency of the observed event. The conception of a
probability finds its justification in the close relation between the
mathematical probabilities and relative frequencies as determined in
a purely empirical way. This relation is established by means
of the famous Law of Large Nimibers " (A. A. Tschuprow).
To return to our original definition of a mathematical proba-
bility as the ratio of the favorable to the coordinated equally
possible cases, we first notice that this definition is wholly
arbitrary like many mathematical definitions. The contention
of Stuart Mill that every definition contains an axiom is rather
far stretched. In mathematics a definition does not necessarily
need to be metaphysical. A striking example is offered in
mechanics by the definitions of force as given by Lagrange and
Kirchhoff. What is force? " Force," Lagrange says, " is a
cause which tends to produce motion." Kirchhoff on the other
hand tells us that force is the product of mass and acceleration.
Lagrange's definition is wholly metaphysical. Whenever a
definition is to be of use in a purely exact science such as mathe-
matics, it must teach us how to measure the particular phe-
nomena which we are investigating. Thus, to quote Poincare,
82
51 ] A PRIORI AND EMPIRICAL PROBABILITIES. 83
" it is not necessary that the definition tells us what force really
is, whether it is a cause or the effect of motion."
An analogous case is offered in the criticism of a mathematical
probability as defined by Laplace, and the attempts to place
the whole theory of probabilities on a purely empirical basis by
Stuart Mill, Venn and Chrystal. These writers contend " that
probability is not an attribute of any particular event happening
on any particular occasion. Unless an event can happen, or
be conceived to happen a great many times, there is no sense in
speaking of its probability." The whole attack is directed against
the defiuition of a mathematical probability in a simjle trial
which definition, evidently by the empiricists, is regarded as
having no sense. The word " sense " must evidently be con-
sidered as having a purely metaphysical meaning. In the same
manner Kirchhoff' s definition might be dismissed as having no
sense, since it would seem as difficult to conceive force as a purely
mathematical i)roduct of two factors, mass and acceleration, as
it is to conceive the definition of a mathematical probability
as a ratio.
The metaphysical trend of thought of the above writers is
shown in their various definitions of the probability of an event.
Mill defines it merely as the relative frequency of happenings
inside a large number of trials, and Venn gives a similar defini-
tion, while Chrystal gives the following:
" If, on taking any very large number N out of a series of cases
in which an event, E, is in question, E happens on piV occasions,
the probability of the event, E, is said to be p."
Let us, for a moment, look more closely into these statements.
Any definition, if it bears its name rightly, must mean the same
to all persons. Now, as a matter of fact, the vagueness in a
half metaphorical term like " any very large number " illustrates
its weakness. The question immediately confronts us " what is
a very large number? " Is it 100, 1,000 or perhaps 1,000,000?
A fixed universal standard for the value of N seems out of the
question and the definition — although perhaps readily grasped
in a " general way " — can hardly be said to be happily chosen.
Another, and perfectly rigorous definition, is the following one
given by the Danish astronomer and actuary, T. N. Thiele.
84 THE LAW OF LARGE NUMBERS. [51
Thiele tells us that " common usage " has assifj;ned the word
probability as the name "for the limiting value of the relative
frequency of an event, when the number of observations (trials),
under which the event happens, ai)j)r()ach infinity as a limit."
A similar definition is later on given by the American actuary
R. Henderson, who says: " The numerical measure which has been
universally adopted for the probability of an event under given
circumstances is the idtimate value, as the number of cases is
indefinitely increased, of the ratio of the number of times the
event happens under those circumstances to the total possible
number of times." There is nothing ambiguous or vague in these
definitions. Infinity, taken in a purely quantitative sense, has a
perfectly uniform meaning in mathematics. The new definition
differs, however, radically from our customary definition of a
mathematical a priori probability. We cannot, therefore, agree
with Mr. Henderson when he continues " the measure there given
has been universally adopted and this holds true in spite of the
fact that the rule has been stated in ways which on their face differ
widely from that above given. The one most commonly given
is that if an event can happen in a ways and fail in b ways all of
w'hich are equally likely, the probability of the event is the ratio
of a to the sum of a and b. It is readily seen that if we read
into this statement the meaning of the words " equally likely," this
measure, so far as it goes, reduces to a particular case of that given
above."
In order to investigate this statement somewhat more closely,
let us try to measure the probability of throwing head with an
ordinary coin by both our old definition of a mathematical
probability and the definition by Mr. Henderson of what we
shall term an empirical probability. Denoting the first kind of
probability by P{E) and the second by P'{E) we have in ordinary
symbols
PiE) = h
P\E) = lim F(E, v)
11=00
where the symbol F{E, v) denotes the relative frequency of the
event, E, in v total trials. No a priori knowledge will tell us
offhand if P'{E) will approach | as its ultimate value. The
52] EXTENT AND USAGE OF BOTH METHODS. 85
two methods are radically different. By the first method tlie
determination of the mimerical measure of a probability depends
simi)ly on our ability to jud^n' and sej^rej^ate the equally possi})le
eases into cases faAorable and unfavorable to the event E. By
tlie second method the (h^termination of the ])rol)ability de])ends,
not alone on the segref:;ation and consequent enumeration of the
favorable from the total cases, but chiefly on the extent of our
observations or trials on the event in question.
52. Extent and Usage of Both Methods. — Before enterin}^ into
a more detailed discussion of the actual quantitative comparison
of the two methods, it mi<2;ht be of use to compare their various
extent of usai^e. In this res])ect the empirical method is vastly
sui)erior to the a ])riori. A rigorous aj)plication of the a ])riori
method, as far as concrete problems <i;o, is limited to sim])le
games of chance. As soon as we begin to tackle sociological or
economical ])ra(tical problems it leaves us in a helpless state.
If we were to ask about the probability that a certain i)erson
forty years of age would die inside a year, it would be of little use
to try to determine this in an a priori maimer. Even>a ])urely
deductive process, as illustrated by Bayes's Rule in the earlier
chapters, leads to ])arad()xical residts. Our a ])riori knowledge
of the complexes of causes governing death or snrvi\al is so
incomplete that even a qualitative — not to speak of a quanti-
tative — judgment is out of the (piestion. The empiricid n:iethod
shows us at least a way to obtain a measure for the probability
of the event in (juestion. By observing during a period of a yejir
an infinite number of forty-year-old ])ersons of wliom, after an
exhaustive qualitative in\'estigation, we are led to believe that
their present conditions as far as health, social occupation, en-
vironments, etc., are concerned are equally similar, we may by
an enumeration of those who died during the year obtain the
desired ratio as defined V)y P'{E). Of course, ol)servation
an infinite number is ])ractically impossible. An approximate
ratio may be formed by taking a finite, but a large, number
of cases under observation. But how large a number? This
very question leads straightforward to another problem, namely
the quantitative determination of the range of variance between
the approximate ratio and the ideal ultimate ratio as defined bv
86 THE LAW OF LARGE NUMBERS. [52
the relation
P'iE) = lim FiE, v).
Since it is impossible to make an infinite number of observations
we cannot find the exact vahie of the ran<i;e of such variations.
But we may, however, determine the ])robabihty tliat this range
does not exceed a certain fixed quantity, say X, in absohite mag-
nitude. Stated in compact form our problem reduces to the
following form: To determine the probability of the existence
of the following inequality:
lim F(E, v) --\^\
where both a and s are finite numbers. This, to a certain extent,
contains in a nut shell some of the most important problems in
probabilities.
The above problem may be solved in two distinct ways. The
first, and perhaps the most logical way, is by a direct process.
This is the method followed by T. N. Thiele in his " Almindelig
lagttagelseslajre,"^ published in Copenhagen, 1889, a most
original work, which moves along wholly novel lines. Thiele
distinguishes between (1) Actual observation series as recorded
from observation, in other words statistical data. (2) Theoret-
ical observation series giving the conclusions as to the outcome of
future observations and (3) Methodical laws of series where the
number of observations is increased indefinitely. By such a
process, purely a theory of observations, the whole theory of
probability becomes of secondary importance and rests wholly
upon the theory of observed series, a fact thoroughly- emphasized
by Thiele himself. When the author first, in the closing chapters
of his book, makes use of the word probability it is only because
" common usage " has assigned this word as the iiaine for the
ultimate frequency ratio designated by our symbol lim F{E, v).
f=00
The prol)lem may, however, be solved in an indirect way,
which is the one T shall adopt. This method, as first consistently
deduced by Laplace, has for its basis our original definition of a
mathematical a priori prol)abilitN' and may be briefly sketched as
follows: We first of all postulate the existence of an a priori
1 English edition, "Theory of Observations," London, 1905.
53] AVERAGE A PRIORI PROBABILITIES. 87
probability as defined, altli()u<i;h its actual determination, by a
priori knowledj^e, is impossible except in a few cases, as, for
instance, simple S'ln^e^ of chance, drawing balls from urns, etc.
Denoting such a probability by P(E), or p, we next ask, What will
be the expected number, say a, of actual happenings of the event,
E, expressed in terms of .s- and i^, when we make s consecutive
trials instead of a single trial, and what will be the number of
happenings of E when s approaches infinity as its ultimate value?
If such a relation is found between j:), a and s, where /-> is the
unknown quantity, we have also found a means of determining
the value of /; in known quantities. Our next question is —
What is the probability that the absolute value of the difference
between p and the relative frequency of the event as expressed
by the ratio of a to s does not exceed a previously assigned
quantity? Or the probability that
a
X?
Now, as the reader will see later, we shall prove that
lim F{E, v) = P{E) = p.
v=x>
It must, however, be remembered that this result is reached by a
mathematical deduction, based upon the postulate of mathe-
matical probabilities, and not in the manner as suggested in the
above statement by Mr. Henderson.
It is only after having established such purely quantitative
relations that we are entitled to extend the laws of mathematical
probabilities as deduced in the earlier chapters to other problems
than the simple problems of games of chance.
53. Average a Priori Probabilities. — In the previous para-
graphs of this chapter, another important matter is to be noted,
namely the assumption that the complex of causes producing
the event in question remains constant during the repeated
trials (observations), or, stated in other words the mathematical
a priori probability remains constant. Under this limitation
the extension of the laws of mathematical probabilities would
have but a very limited practical application. In all statistical
mass phenomena such an ideal state of affairs is rather a verv
88 THE LAW OF LARGE NUMBERS. [54
rare exception. If we consider an ordinary mortality investiga-
tion we know with absolute certainty that no two persons are
identically alike as far as health, occupation, environment and
numerous other thin(i;s are concerned. Thus the postulated
mathematical probability for death or survival during a whole
calendar year will in general be different for each person. We
may, however, conceive an average probability of survival for a
full year defined by the relation
V\ + /J2 + 7^3 + • • • Ps 2/?
^' = . ~ = T'
where p\, p-i, pz, • • • are the postulated probabilities of each
individual u!ider observation. Our task is now to find:
1. An algebraic relation between the average probability as
defined above, the absolute frequency a and the total number of
observations (trials) s,
2. The same relation when s approaches a as its ultimate value,
3. The probability of the existence of the inequality,
a
^X,
where a denotes the absolute frequency- of the occurrence of the
event, s the total number of observations (trials) and X an ar-
bitrary constant.
54. The Theory of Dispersion. — As we mentioned before the
empirical ratio ajs represents only an ap})roximation of the ideal
ultimate value of lim F{E, r). If we now make a series of
observations (trials) on the occurrence of a certain event E, such
that instead of a single set of observations of s individual ob-
servations we take A^ such sets, we shall have A^ relative frequency
ratios :
,f ' ,V ' ,9 ' S '
Since the ratios are a])])r()ximations only of the ultimate ratio
they will in general exhibit discrepancies as to tlieir numerical
values and may be regarded as A"^ different empirical a]))>roxima-
tions. "^I'he question now arises how these various empirical
ratios group themselves around the value of lim F{E, r). The dis-
55] HISTORICAL DEVELOPMENT OF LAW OF LARGE NUMBERS. 89
tribution of the empirical ratios around the ultimate ratio is by-
Lexis called " dispersion."
55. Historical Development of the Law of Large Numbers. —
The first mathematician to investigate the problems we have
roughly outlined in the previous paragraphs was the renowned
Jacob Bernoulli in the classic, " Ars Conjectandi," which rightly
may be classified as one of the most important contributions on
the subject. Bernoulli's researches culminate in the theorem
which bears his name and forms the corner-stone of modern
mathematical statistics. That Bernoulli fully realized the great
practical importance of these investigations is proven by the
heading of the fourth part of his book which runs as follows:
" Artis Conjectandi Pars Quarta, tradens usum et applicationem
praecedentis doctrinae in civilibus et oeconomicis." It is also
here that we first encounter the terms " a priori " and " a pos-
teriori " probabilities. Bernoulli's researches were limited to
such cases where the a priori probabilities remained constant
during the series or the whole sets of series of observations.
Poisson, a French mathematician, treated later in a series of
memoirs the more general case where the a priori probabilities
varied with each individual trial. He also introduced the technical
term, " Law of Large Numbers " (" Loi des Grand Xombres ").
Finally Lexis through the publication in 1877 of his brochure,
" Zur Theorie der ]Massenerscheinungen der menschlichen Gesell-
schaft," treated the dispersion theory and forged the closing
link of the chain connecting the theory of a priori probabilities
and empirical frequency ratios. Of late years the Russian mathe-
matician, Tchebycheff, the Scandinavian statisticians, Wester-
gaard and Charlier, and the Italian scholar, Pizetti, have con-
tributed several important papers. It is on the basis of these
papers that the following mathematical treatment is founded.
In certain cases, however, we shall not attempt to enter too
deeply into the theory of certain definite integrals, which is
essential for a rigorous mathematical analysis, but which also
requires an extensive mathematical knowledge which many of
my readers, perhaps, do not possess. To readers interested in
the analysis of the various integrals we may refer to the original
works of Czuber and Charlier.
CHAPTER VIII.
INTRODUCTORY FORMULAS FROM THE INFINITESIMAL
CALCULUS.
56. Special Integrals. — In the following chapters we shall
attempt to investigate the theory of probabilities from the stand-
point of the calculus. Althougli a knowledge of the elements
of this branch of mathematics is presupposed to be possessed
by the student, we shall for the sake of convenience briefly
review and demonstrate a few formulas from the higher analysis
of which we shall make frecjuent use in the following paragraphs.
All such formulas have been given in the elementary instruction
of the calculus, and only such readers who do not have this
particular branch of mathematics fresh in memory from their
school days need pay any serious attention to the first few
paragraphs.
57. Wallis's Expression for tt as an Infinite Product. — We wish
first of all to determine the value of the definite integral:
Jn = X'%in-rrf.r, (1)
under the assumption that n is a positive integral number. This
integral is geometrically equal to the area between the x axis,
the axis of y, the ordinate corresponding to the abscissa \tv and
the graph of the function y = sin" .r. Letting // = D^n = sin x,
V = sin"~^ X, we get by partial integration:
J„ = _ cos .r sin"-i .r] ^ '--\- J" '~ cos .r(/i- 1) sin"-^ x cos xdx. (2)
If we substitute the upper and lower limits in the first term on
the right hand side of the above expression for J„ this term
reduces to 0, assuming /( > 1. Thus we have:
Jn = in — D^J*"^" sin"~2.r-cos2a:c?a:.
90
57] WALLIS'S EXPRESSION OF tt AS AN INFINITE PRODUCT. 91
Putting cos^ .T = 1 — sin^ x, we get:
Jn = (/I - l)o/"^" sin"~- xdx - {n - l)J^'^' sin" xdx. (3)
The last integral is, however, equal to J„ and the first integral
is, following the notation from (1), equal to J„_2. We shall
therefore have:
Jn + in — \)Jn = (n — l)e/„_2,
or
nJn = (W - 1) Jn-2. (4)
Replacing nhy n — 1, n — 2, n — 3, • • • successively we get:
nJn = (W — 1)J„_2,
(n - l)Jn-i = {n - 2)J„_3,
(n — 2)J„_2 = (w — 3)J"„_4,
According as n is even or uneven we shall have one of the
following equations at the bottom of the recursion formula:
Jo = oj " sin° xdx = ^^ " dx = Jx,
or
Ji = oJ " " sin xdx = — cos a- T " = 1. (5)
If, for even values of 7i, we let n = 2m, and, for uneven values,
n = 2m — 1, we get finally the following recursion formulas:
2W J2m = (2 m — l)J2m-2, (2m— l)J-2m-l = (2 W — 2)J2m-Z,
{2m - 2)J2n.-2= (27w-3)J2^_4, (2m-3)J2^_3= (2m-4)J2^_5,
2J2 = l47r, 3/3=2X1.
Successive multiplication of the above equations gives us
finally :
^ (2m- l)(2m-3)-»-l tt
''^"^ 2m(2m-2)---2 ^2'
_ (2m - 2)(2m - 4)>--2 ^^^
/2m-i - ^2vi- l)(2m- 3)--.3'
We may now draw some very interesting conclusions from the
92 FORMULAS FROM THE INFINITESIAL^L CALCULUS. [58
above equations. Both integrals represent geometrically areas
bounded by the graphs of the functions:
y = sin-'" .r and y = sin-"*~^ .r respectively.
The difference of the ordinates of these graphs, namely:
(sin .r — 1) sin-"'-! x
is evidently decreasing with increasing values of the positive
integer n, since sin x lies between and + 1 and sin-'""^ x ap-
proaches the value except for certain values of .r. The larger
we select ?// the less is the difference of the two areas and the
ratio will therefore approach 1, or the expression
{2m- 2)(2m - 4)---2 ^ {2m - l)(2m - 3) •••3 _ tt
(2m- l)(2w - 3)---3 ' 2m(2w-2)---2 ~ 2'
Hence:
TT
lim
22.42.62...(2to- 2)2. 2m
2 ;,:r; l- • 3-^ • S^ • • • (2m - 3)2(2m - 1)2
Multiplying with 2--4--62- • •{2m — 2)- we get:
X ,. 2'"'-h7i[{m - 1)/]^ ,. 22'"(m/)2
L.= ^J7/2.
- = am — rTT, TTT^T^ — or i:m —
2 „,=« [{2m - 1)/]- „,^« (2,„/) M2m
This is the formula originally discovered by the English
mathematician, John Wallis (1616-1703), and by means of which
TT may be expressed as an infinite product.
58. De Moivre — Stirling's Formula. — ^We are now in a position
to give a demonstration of Stirhng's formula for the approximate
value of n! for large values of n. A. de Moivre seems to have
been the first to attempt this approximation. In the first edition
of his "Doctrine of Chances" (1718) he reaches a result, which
must be regarded as final, except for the determination of an
unknown constant factor. Stirling succeeded in completing this
last step in his remarkable "Methodus Differentialis" (1738).
In the second edition of "Doctrine of Chances" (1738) de Moivre
gives the complete formula with full credit to Stirling. He
mentions as his belief that Stirling in his final calculation possibly
has made use of the formula of Wallis. The demonstration by
the older English authors is rather lengthy and much shorter
58] DE MOivRE — Stirling's formula. 93
methods have been devised by later writers. ]\I()st authors
make use of the Euk-riau integral of the second order by which
any factorial may be expressed by a gamma function:
r(/t + 1) = J^^x^e-^dx = n!.
Another method makes use of the well-known Euler's Summation
Formula from the calculus of finite differences. This metliod is
of special interest to actuarial students, who frequently use the
Eulerian formula in the computation of various life contingencies.
For the benefit of those interested in this particular method we
may refer to the treatises of Seliwanoff and Markhoff, two
Russian mathematicians.^
The Italian mathematician, Cesaro, has, however, derived
the formula in a nuich simpler manner.-
Cesaro starts with the inequalities:
1
(-;r'
1+
e < 1 + - <e '^"*"+'^
From a well-known theorem from logarithms we have:
2 ^^^' n 2n + 1 "^ 3(2n + l)^ "^ 5(2n + 1)^ "^ ' * ''
which also may be written as follows:
,V = (n + \) log. (|l + )J= 1 + 3(2nVl)^"'"5(2^W+- ' •'
If all the coefficients 3, 5, • • • are replaced by the number 3,
we obtain a geometrical series. The summation of this infinite
series shows that
1 < .V < 1 -I
/ l\n+l/2 ,, 1
or
If we let
_ nXe"" _ {n -f l)!e"+i
'Seliwanoff, " Lehrbuch dor Differenzenrechnung," Leipzig, 1905, pages
59-60; Markhoff, "Differenzenrechnung," Leipzig, 1898.
'^ Cesaro, "Corso di analisa algebrica," Torino, 1884, pages 270 and 480.
94 FORMULAS FROM THE INFINITESIMAL CALCULUS. [58
then
Wn+1 e
Dividing the quantities in (I) by e we have:
Un
1 <_Ii-< ^12«(« + 1)^ (II)
The exponent of e may be written as follows:
1 1 1
12n{n + 1) 12n 12(w + 1) *
^Making use of this relation (II) may be written in the following
form:
Denoting the quantity: Un-e~^'^-" by m„', we shall have two mon-
otone number sequences:
Ul, U-2, UZy' ' ' ^(n, Wn+1, ' ' ' ,
111', n-2, Ih',- ■ -Un, Un+U ' ' ••
These two sequences show some very remarkable features.
With increasing values of n the values of u„ decrease, or the
sequence is a monotone decreasing number sequence. The
values of ?/,/ become larger when n is increased and form there-
fore a monotone increasing number sequence. But any member
of this latter series satisfies, however, the inequality
Un' < Un.
Since both number sequences are situated in a finite interval
it follows from the well-known theorem of Weierstrass that they
both have a clustering point, i. e., a point in whose immediate
region an infinite number of points of the sequence are located.
Denoting this point of cluster by a, we have here an increasing
and a decreasing monotone sequence which both converge
towards a, or:
lim u,/ = lim tin = a-
n=oo n=oo
This relation may be illustrated by the accompanying diagram:
If we now let lim //„ = lim ?/„-e~^''^^" = o, then we shall have
58 J DE MOiVRE — Stirling's formula.
for every finite value of 71 :
where a = i/„-c-^'i2" (() < ^ < i)
95
^ ^i^ z 3 4
This gives us finally the following expression for /;!:
ail)
In this expression we need only determine the unknown
coefficient a. The formula of Wallis gives immediately:
,. (2-4-6---2/?02 ,. 22"(n!)2 ,^
hm ,— -" = lim — — — =^ = "V7r/2.
(2w) ! V2n n=oo (2w) ! V2?
Substituting in this latter expression the value for factorials
as found in (III) and neglecting the quantity: djV2n, we have
after a few reductions:
lim
an
= V^/o
=00 ^2'n{2n)
7r/2, or a = Vi^Tr,
from which we easily obtain De ]\Ioivre-Stirling's Formula in its
final form:
n\ = V27r-n"+^/2-e-".
This remarkable api^roximation formula gives even for com-
paratively small values of n surprisingly accurate results. Thus
for instance we have:
10! = 3,628,800; Wh-'''^2Q-K = 3,598,699.
CHArXER TX.
LAW OF LARGE NUMBERS. MATHEMATICAL DEDUCTION.
59. Repeated Trials. — Let us consider a general domain of
action wherein the determining causes remain constant and
produce either one or the other of the opposite and mutually
exclusive events, E and E, with the respective a priori prob-
abilities p and q (q = 1 — p) in a single trial. The trial (observa-
tion) will, however, be repeated s times with the explicit assump-
tion that the outward conditions influencing the different trials
remain unaltered during each obser^'ati()n. The simplest ex-
ample of observations of this kind is offered by repeated draw^ings
of balls from an urn containing white and black balls only, and
where the ball is put back in the urn and mixed thoroughly with
the rest before the next drawing takes phice. We keep now a
record of the repetitions of the opposite events, E and E during
the s trials, irrespective of the order in which these two events
may happen. This record must necessarily be of one of the
following forms:
E happens s times, E times,
E " s - 1 " E 1 "
E " s-2 " E2 "
E " " Es
In Chapter IV, Example 17, we showed that the probabilities
of the above combinations of the two events, E and E, were
determined by the expansion of the binomial
(p + <])"•
96
61] SIMPLE NUMERICAL EXAMPLES. 97
The general term
is the probabiHty P{E'^E^) that E will happen a and /•; /3 times
in the s total trials. Each separate term of the binomial expan-
sion of {p + qY, represents the probability of the happening (^f
the two events in the order given in the above scheme.
60. Most Probable Value.— In dealing with these various
terms, it has usually been the custom of the P^nglish and French
mathematicians as well as many German scholars to pay par-
ticular attention to a special term, the maximum term, which
generally is known as the "most probable value" or the "mode."
Russian and Scandinavian writers and the followers of the Lexis
statistical school of Germany have preferred to make another
quantity known as tlie "probable" or "expected value," the
nucleus of their investigations. Although it is our intention to
follow the latter method, we shall discuss first, briefly, the most
probable value. Two questions are then of special interest
to us:
(1) WTiat particular event is most probable to happen?
(2) What is the probability that an event will occur whose
probability does not differ from that of the most probable event
by more than a previously fixed quantity?
Neither of the two questions offers any particular principal
difficulties from a theoretical point of view. When regarding
the probability P{E'^E^), which we shall denote by T, as a func-
tion of the variable quantity, a, T evidently will reach a maximum
value for a certain value of a, {^ = s — a), and we need only
determine the greatest term in the above binomial expansion.
In order to answer the second question we have only to pick
out all the terms which are situated between the two fixed limits.
Their sum is then the probability that those two limits are not
exceeded.
61. Simple Numerical Examples. — ^^^len * is a comparatively
small number the actual expansion may be performed by simple
arithmetic. We shall, for the benefit of the student, give a
simple example of this kind.
8
98 LAW OF LARGE NUMBERS. [61
A pair of dice is thrown 4 times in succession, to investigate
the chance of throwing (h)ublets.
In a single throw the probabiHty of getting a doublet is
V ^ r A 9 — r ) • Expanding ( 7^ + 7^ ) by means of the bi-
nommal theorem we get l-rl +"^17^) \r/''b\r) \c I
~^^\r)\r) ~^ \ r ) ' ^^^^ ^^ ^^^^ above terms represents
the ])r()l)ability of the occurrence of the various combinations of
doublets (E) and non-doublets (E), and it is readily seen that
the event of getting no doublets at all, represented by the
o^^
1-ist term ( - ) = ('•4823, has the greatest probability. In other
w trils it is the most probable event.
Let us next repeat the trial 12 times instead of 4. The 13
possible probabilities for the various combinations of doublets
and non-doublets will then be expressed by the respective terms
in the expression
{I^IT-
The 13 members have as their common denominator the quantity
2,176,782,336 and as numerators the following quantities: 1, 60,
1,650, 27,500, 309,375, 2,475,000, 14,4:57,500, 61,875,000, 193,-
359,375, 429,687,500, 644,531,250, 585,937,500, 244,140,625,
which now shows that the most probable combination is the one
of 2 doublets and 10 non-doublets, having a numerical value
equal to .2961.
A further comparison will show that the most probable
event in the second series had the pr<)l)al)inty .29()1, whereas
.4823 was its value in the first series. In other words, the prob-
ability decreases when the trials (observations) are increased.
This is due to the fact that the total number of possible cases
becomes large with the increase of experiments.
Another question which presents itself, in this connection,
is the following: What is the probability that an event will occur
62] VALUE IN A SERIES OF REPEATED TRIALS. 99
whose probability does not differ from the most probable value by
more than a previously fixed quantity? Let us suppose we were
asked to determine tlie probability that a doublet does not occur
oftener than 5 times and not less than 1 time in 12 trials. This
probability is found by adding the numerical values of the prob-
abilities as given in the binomial expansion from the term
containing i> = tt to the power 6, to p to the first power or
14,437,500 + 0,187,500 + 193,359,375 + 429,087,500
+ 044,531,250 + 585,937,500
2,170,782,336
62. The Most Probable Value in a Series of Repeated Trials.
— In the examples just given we determined the probability for
the happening of the most probable event in a series of s observa-
tions by a direct expansion of the binomial (p -\-q )*. This
may be done whenever s is a comparatively small number. But,
when s takes on large values, this method becomes impracticable,
not to say impossible. Suppose that s — 1,400, then the actual
straightforward expansion {p + r/)^"*"'^ would require a tremen-
dous work of calculation which no practical computer would be
willing to undertake. We must therefore in some way or other
seek a method of approximation by which this labor of calcula-
tion may be avoided and try to find an a])proximate formula by
which we are able to express the maximum term in a simple
manner, involving little computation and at the same time
yielding results close enough for practical as well as theoretical
purposes. Jacob Bernoulli in his famous treatise "Ars Conjec-
tandi" was the first mathematician to solve this problem.
Bernoulli also gave an expression for the probability that the
departure from the most probable value should not exceed pre-
viously fixed limits. The method, however, was very laborious
and the final form was first reached by Laplace in "Theorie des
Probabilites."
We saw before in Chapter IV that the general term
100 LAW OF LARGE NUMBERS. [ G2
ill the binomial exi^ansion (p + 7)* represented the probabiHty
that an event, E, will hap])en a times and fail /3 times in s trials,
where p and q were the respective probabilities for snceess and
failure in a siiiji;le trial. The exponent a may here take all posi-
tive integral values in the interval (0, s), including both limits.
The question now arises, which particular value of a, say «„,
will make the above quantity a maximum term in the expansion
of the binomial? If a,, really is this ])articular value, then it
must satisfy the following inequalities:
gl gl
(I) (11)
= (an- l)!(^n+l)!^ ^ •
(HI)
Dividing (II) by (III) and (II) by (I) we obtain the following
inequalities:
an q - I3n p -
which also may be written
(/3„ + l)p ^ qan and («„ + l)q ^ ^np.
The following reductions are self evident:
(s — a„ + l)p ^ a:„(l — 2^) or sp -\- p "^ an,
and
{an + l)q ^ {s — an)p or a,// + anp '^ sp — q or a„ ^ sp — q.
From which we see that «„ satisfies the following relation:
ps — q ^ an ^ ps -\- p.
Since p -{- q = 1, we notice that an is enclosed between two
limits whose difference in absolute magnitude equals imity.
The whole interval in which a„ is situated being equal to unity,
and since a„ must b(> an integral number, this particular a„ is
deteniiiiHil iiiii(|U('ly as an integral positive number when both
ps — q and /AS- + p are fractional quantities. Tf ps — 7 is an
integral number ps + p will also be integral, and «„ had to be a
63] APPROXIMATE CALCULATION OF THE MAXIMUM TERM. 101
fractional number in order to satisfy the above inequality.
Since by tlie nature oF the probk^m a,, can take j)ositive integral
values only, the binomial expansion of {p + q)^ must have two
terms which are greater than any of the rest. Dividing both
sides of the inequality by s, we shall have
V < — <7)+-or(7 + -> — and « + - > ^ .
s s ^ s ^ s .y ^ s s
Since botli p and q are proper fractions, both />/.s' and qls are less
than 1/6". We may therefore safely assume that the highest pos-
sible difference between the two quotients cv„/,s' and ^n/s and the
probabilities p and q will never exeeed 1 .9. Now if s is a very
large number this quantity may be neglected, and we may
therefore write ps = an and qs = j3n.
Substituting these values in our original expression for the
general term of the binomial expansion we get as the maximum
number:
63. Approximate Calculation of the Maximum Term, T^. —
When the trials are repeated a large number of times the straight-
forward calculation of the maximum term becomes very laborious.
The only table facilitating an exact computation is in a work
"Tabularum ad Faciliorem et Breviorem Probabilitatis Com-
putationem Utilium, Knneas," by the Danish mathematician,
C. F. Degen, This table, which was published in 1824, gives the
logarithms to tweh'c places for all values of nl from n = 1 to
n = 1,200. Degen's table is, however, not easily obtained, and
even if it were, it would be of little or no value for factorials
above 1,200 !. Our only resort is therefore to find an approximate
expression for the above value of n !. This is most conveniently
done by making use of Stirling's formula for factorials of high
orders. We have
s\ = 5^i\-*V27,
{sq)\ = {sqy'^+'^'^e-''^^l2^.
102 LAW OF LARGE NUMBERS. [64
Substituting the above values iu the expression s!/{(,sp)! (sq)!)
we get
1
Hence we have
which reduces to
T =
njSPqSq
jjSp+1 n^sq+l /2 -^2x5 '
1
T =
X2ir.spq
as an approximate value for the maximum term.
Tchehychcff\s Theorems. — Despite all that has been said about
the most probable value, its use is somewhat limited, and it
might well, without harm, be left out of the whole theory of
probabilities. Just because an event is the most probable it
does by no means follow it is a very i)robable event. In fact the
expression ( '^'2-Kspq)~'^ which for large values of s converges
towards zero shows that the most probable event in reality is a
very improbable event. This statement may seem a little
paradoxical; but it is easily understood by realizing that the
most probable event is only a probability for a possible combina-
tion among a large number of equally possible combinations of a
different order.
Instead of finding the most probable event it is more important
in practical calculations to determine the average number or
mean value of the absolute frequencies of successes. In Chapter
V we pointed out the close relation between a mathematical
expectation and the mean value of a varial)l('. This relation is
used by the Russian mathematician, Tchebycheft", as the basis
of some very general and far-reaching theorems in probabilities,
by means of which the Law of Large Numbers may be established
in an elegant and elementary manner.
64. Expected or Probable Value. — In Chapter V we defined
the product of a certain sum, s, and the probabiiitN' of winning
such a sum as the mathematical expectation of .v. It is, however,
not necessary to associate the happening of the event with a
monetary gain or loss, in fact it serves often to confuse the
reader and we may generalize the definition as follows. 7/ a
64] EXPECTED OR PROBABLE VALUE. 103
variable at may assninc any of the values ai, a-i, az • • • a^ each with
a resjjective probability uf existence (p{ai) (i = 1,2 • • • s) and such
that X(p{ai) — 1, then we define:
llanpiai) = e{ai)
as the expected value of a i.
Some writers use also the term probable value instead of
expected value. In other words the expected value of a variable
quantity, a, which may assume any one of the values ai, a-i- • •««
is the sum of the products of each individual value of the variable
and the corresponding ])ro})ability of existence of sucli value.
Suppose we now have two o])posite and complementary events
E and E for which the probabilities of happening in a single
trial are equal to p and q = 1 — p respectively. When the
trials are repeated s times the probabilities of E hapi)ening s
times, E no times, of E hai)])ening s — I and E once, of E s — 2
and E 2 times and so on, may be expressed by the individual
terms of the expansion:
(p + qy,
where the general term expressing the occurrence oi E a times
and of E (s — a) times is:
which is also the pio})ability of the existence of the frequency
number a. The variable in the binomial expansion is a, which
may assume all values from to .s- inclusive.
We now first of all proceed to find the expected value — or the
mathematical expectation — of the following quantities:
a, [a — e{a)] and [a — e(a)p.
We shall presently show the reason for the selection of the
abo^'e exi)ressi()ns, which perhaps may appear at the present,
somewhat i)uzzling to the student.
In mathematical symbols the expected values of the abo\'e
quantities are expressed as follows*
e(a) = Zaifia), e[a — e(a)] = Z[a — e(a)]if(a)
and
e[a - e(a)Y = ^[a - e(a)]V(«)
and the summation is to take place from a = and to a = *.
104 LAW OF LARCiE NUMBERS. [65
65. Summation Method of Laplace. The Mean Error. —
The analytical difficulty lies in the summation of the expressions
as given above. Laplace was the first to give a compact expres-
sion for the different sums in a simple and elegant manner.
By the introduction of the parameter / Laplace writes:
<p(a) = (p + qy = ^ (^'^^ p'^q^'^
as
(p{ta) = (tp+ qY = ^y^j (fp)''q'-\
Differentiating with respect to /, which it must be remembered is
introduced as an auxiliary parameter only, we have:
<p'(ta) = sp{tp + qy-' = -«/M M (tpy-'q'"'.
Letting t assume the special value 1 the above sum becomes e (a)
or
e{a) =Za f M /A/^-« = sp(p + 7)^1 = sp. (L)
We might, however, have obtained the same result in a much
shorter manner by the following consideration. The expecta-
tion for a single event among the s events is equal to p. Since
all the events are independent of each other, it follows from the
addition theorem that the complete expectation of the total s
cases is equal to sp.
We next proceed to determine the expression: c[a — e(a)] or
the expected value of the differences between the constant,
e(a) = sp and the individual values 1, 2, 3, • ■ -, s which a may
assume in the binomial expansion.
The difference a — e(a.) is known as the departure or devia-
tion from the expected value, some of these deviations will be
positive, namely all the values situated to the right of the maxi-
mum term, which also is the most probable term in the expansion
(p + q)", while the a's situated to the left of the maximum value
of a will be less in magnitude than the largest a — sp and the
deviation will therefore be negative. On account of the sym-
metrical form of the binomial expansion we may expect an
Co] SUMMATION METHOD OF LAPLACE. 105
equal number of positive and ne<;ative deviations which, taken
two and two at a time, are etjual in absolute magnitude. The
algebraic sum of all the deviations may therefore be expected
to be equal to zero. We shall, however, in a rigidly analytical
manner prove that this is actually so. We have
e[a — e(a)] = Z[a — e{a)](p{a) = Za(p(a) — ^e{a)(p{a)
= Xa(p(a) — sp2,(p{a.).
The first term in tliis last expression we found, however, to be
equal to e{a) = sp, and we liave finally
e[a — e{a)] = sp — sp = 0.
By squaring the quantity, a — eia), we get a- — 2ae{a) +
[e(a)]-, which is always positive no matter if the above difterence
is negative.
As a preliminary step we shall find
e{a-) — 1,orip{a).
Introducing the auxiliary parameter, /, we g,et:
C)
-'"('.)
{ipYq^- = itp + qY.
The first derivative with respect to t is:
{tpr-'cr" = spitp + 7)<-i.
Multiplying both sides of the equation by //:*, we have:
2pa(/p)Y'-"(^) = stp'{tp+q)-\
Differentiating we get :
2p-Q- r j ifpY-'q"-^ = sp-(ip + 9)^1 + s(s - \)pH{tp + q)^\
Dividing through with the constant factor p and letting t — 1
we have:
-(:)
p'^q^^ = s-p- -{- sp{\ — p) = s^p- + spq.
The expression on the left side is, however, nothing less than the
algebraic sum of ^crip{a) or simply e{cr). This leaves the final
result :
lOG LAW OF LARGE NUMBERS. [66
e{a-) = s-p- + s])q.
We have now:
[a — ('{a}]- = a~ — '2ae(a) -f- [^(«)]^
from wliic'li it follows:
e[a — i'(a)]- = s-p- + spq — 2s- p- + s-p- = spq.
Denotinii tins latter fiiiantity by the symbol [e(a)]- we have:
[e(a)]- = <'[cx — ('{a)]- = spq, or e(a) = ^lspq. (II.)
The quantity e{a) or sim])ly e is commonly known as the mean
error of the frequency number a in the Bernoullian expansion.
The mean error is one of the most useful functions in the theory
of ])robabilities and furnislu\s oiu^ of the most powerful tools of
the statistician.
66. Mean Error of Various Algebraic Expressions. — We next
proceed to prove some general theorems connected with the
mean error. The mean error of the sum of two observed vari-
ables, a and /3, is given by the formula:
e(a -h(3) = Ve2(a) + e^-
Proof: Let e(a) = Za<p(a) and ci(3) = Z/3i/^(/3)
6^(a) = Z[a - e{a)T<p{a) and 6^(^) = Z[/3 - K/3)]V(/5)
be the respective expressions for the probable \alues and the
mean errors of a and /3 where of course 2</?(a!) = 1 and 22i/'(jS) = 1.
Now (p{a^) is the ])robability for the occurrence of the special
value a^, of the variable values, in the same way i/'(/?^) is the
probability for the occurrence of /?^. If cv and /3 are independent
of each other, then according to the multiplication theorem,
<f(ay)\p(^,j^) rei)resents the probability for the simultaneous
occurrence of a^ and /3^ as well as the probability of the occurrence
of the difference: a^ -{- (3^ — e{a) — r(^), since the i)r()bable
values e{a) and e{j3) are constant quantities independent of
either a or |3.
If e denotes the mean error of a + |3 then it follows from the
definition of e that e- = :i:Z[a + /3 - da) - e(i3)]V(«)iA(/3) where
the double summation is to take place for all possible values of
the variables a and (3.
The above expression may be written as:
66] MEAN EKKOK OK VARIOUS ALGEBRAIC EXPRESSIONS. 107
or
A mere inspection will satisfy that the first and the last terms of
this expression equals e^{a) and e-(fi) respectively. The first
term may be written as follows:
since Z\J/(^) = 1. The same also holds true for the last term.
With regard to the middle term we found before that
e[a — e{a)] = 0.
Hence it follows by mere inspection that this term becomes 0.
Thus we finally have:
e\a + /?) = e\a) + eHl3) or e(a + /3) = ^le'(0) + e'(a).
Since the middle term is always 0, it follows a fortiori
e(a - /3) = V6--'(a) + e\^),
also that
€(ka) = ke(a),
where k is a constant. This gives us the following theorems:
The mean error of tlie sum or of the difference of two quantities
is equal to the square root of the sum of the squares of each
separate mean error. The mean error of any quantity multiplied
by a constant is equal to this same constant multiplied by the
mean error of the quantity. (See Appendix.)
The above theorems may easily be extended to any number of
variables: a, (3, y ■ • • so that in general we have
e(« + |3 + 7- • •) = Ve» + 62(^) + 6^(7) + ....
We shall later make use of this formula by a comparison of
the different rates of mortality among different population groups.
So far we have computed the mean error for the absolute
frequencies of a, and the quantity '\spq was compared with the
most probable number of successes sp. But it may also be useful
to know the mean error of the relative frequencies. This calcula-
tion is performed by reducing the mean error of the absolute
108 LAW OF LARGE NUMBERS. [ G7
frequencies to the same degree as these absohite frequencies are
reduced to relative frequencies. We saw before that e(a) = sp.
The rehitive frequency of the probable value is eiaj/s = spjs = p.
The mean error of p therefore is
e[e{cx) -.s]
\p(l
The following remarks of Westergaard are worthy of note:
"When a length is measured in meters and this measure may be
effected with an uncertainty of say 2 meters, the length in
centimetres is then simjily found by multiplication by 100 and
the uncertainty is 200 cm. When we wish to find the mean
error of p instead of sp we only need to divide the mean error
■yJspq hy s, wliich gives -^pq/s.'"
The same result is also easily obtained from the formula
t{l-ci) = ke{a)
when we let A* = 1 s.
67. Tchebycheff's Theorem.— Tcliebychef^"s brochure ap-
peared first in Liouville's Journal for 1866 under the title "Des
valeurs Moyennes." A later demonstration was given by the
Italian mathematician, Pizetti, in the annals of the University
of Geneva for 1892. The nucleus in both Tchebycheff's and
Pizetti's investigations is the expression for the mean error:
^a) = ^[^ - K^)]Va). (1)
The variable ^ may be of any form whatsoever, it may thus for
instance be the sum of several variables: a, ^, y • • • while (p{^)
is the ordinary probability function for the occurrence of ^. Let
us denote the difl'erence: ^r — ^(^r) by iv(r = 1, 2, 3 • • • s). We
may then write the above expression for e{^) as:
<^(^l) ^ + ^{h) V2 + <^(^3) 7! + • • • <p{is) ^' = "~^^ (2)
It (J/ CV (/ (*
where a is an arbitrarily chosen constant, but always larger than
e{^) in absolute magnitude. If we, in the above equation, select
all the r's which are larger than a in absolute magnitude together
with their corresponding probabilities, <piO ^^d denote all
»2
> 1
68] tchebycheff's theorem. 109
such quantities by v', v", v"\ ••• and <p{0\ <p{0" , <p{^)"', ■■'
respectively, we have evidently:
a" "^ a^ "^ a2 "^ " * ^ a^ ^'^^
For any one of these different v's which is larger in absolute
magnitude than a
from which it follows a fortiori:
<pi^y + <piO" + ■'■ = ^^KO < ^ . (3a)
In this latter inequality, "^(p^^) is the total probability for the
occurrence of a deviation from e{^) larger than a in absolute
magnitude.
Let now Pt be the probability that the absolute value of
the mean error is not larger than a; then 1 — Pt is the total
probability that the mean error is larger than a. We have thus
from the inequality (3a)
1 — Pt< — 5- or Py > 1 5- • (4)
Let also a — Xe(^). We then have by a mere substitution in the
above inequality:
Pt>1-^,. (5)
This constitutes the first of Tchebycheff's criterions which says:
The probability that the absolute value of the difference \ a — e{a) |
does not exceed the mean error by a certain multiple, X, (X > 1) is
greater than 1 — (l/X-).
Now we made no restrictions as to the variable, ^, which may
be composed of the sum of several independent variables, a, /3,
7, • • • . We saw before that
e\a + ^ + 7 + • • •) = 62(a) + e\^) + 6^(7) + • • •
Tchebycheff's criterion may therefore be extended as follows:
The Tchebycheffian probability, Pt, that the difference | a + /3 + 7
-\- • ■ • — e{a) — e{^) — e{y) — • • • 1 loill never exceed the mean
error thy a certain midtiple, \> \, is greater than 1 — (l/X^).
110 LAW OF LARGE NUMBERS. [69
68. The Theorems of Poisson and Bernoulli proved by the
application of the Tchebycheffian Criterion. — Bernoulli in his
researches limited himself to the solution of the problem in which
the probabilities for the observed event remained constant during
the total luiniber of observations or trials. Poisson has treated
the more general case, wherein tlie individual probal)ility for the
happening of the event in a single trial varies during the total s
trials. This may probably best be illustrated by an urn schema.
Suppose we have s urns Ui, U2, ■ " Us with white and black
balls in various numbers. Let the probability for drawing a
white ball from the urns Vi, Uo, ••• Us in a single trial be
Pi, Ih, ■ • • Ps respectively, r/i, r/o, • • • q, the chances for drawing
a black ball in a single trial. If a ball is drawn from each urn,
what is the probability of a drawing a white and .v — a black
balls in s trials? It is easily seen that the Bernoullian Theorem
is a special case when the contents of the s urns and the respective
probabilities for drawing a white ball in a single trial are the
same for all urns.
69. Bernoullian Scheme. — We shall now show how the Tche-
bycheffian critierions may be used in answering the question
given above. First of all we shall start witli tlie simi)ler case
of the Bernoullian urn-schema. Here the probability for drawing
a white or a black ball from each of the s urns in a single trial is
J) and q respectively. The square of the mean error in a single
trial is pq. From the formulas in § GG it then follows:
e- = er + €2" + • • • = PQ -\- P'l + ;'7 + * ' ' -^ times = spq
or _
e = ^l.S'pq.
While the above expression gives us the mean error of the absolute
frequency of the variable a, the relative frequency of a to the
total number of trials, s, is given as
^fpq
We now ask: WTiat is the total probability that the absolute
deviation of the relative; frequency a/s from its expected value
sp/s = p never becomes larger than X times the mean error.
70] poisson's scheme. Ill
c — 'Spq/s? Letting X = 'Sslt and using the symbols Pr for
this particular probability, we have according to Tchebycheff's
criterion :
Pt> 1 - 1/X', or Pt > 1 - f/s.
Since the mean error is equal to yxjis we have:
The answer to our question above follows now a fortiori as
follows :
The total probability that the absolute deviation of the relative
frequency from the postulated a priori probability, j), never
exceeds the fpiantity, '\pqlt, is greater than 1 — (^-/.v).
By taking / large enough we may reduce ^IjH/lt (where pq is
a fraction whose maximum value never can exceed 1 -r- 4,) below
any previously assigned quantity, 5, however small. If, for
instance, we choose the value .0001 for 5, we may rest assured
that 'ylpq/t will be less than 5 when we take / larger than 5000.
But no matter how large t is, so long as it remains a finite number,
by letting s = cc as a limiting value, fjs will simultaneously
approach as a limiting value. From the deductions thus
derived we are now able to draw the following conclusions:
1) By letting s = oo as a limiting value, the probability, Pt,
that the absolute difference between the relative frequency ajs and the
postulated a priori probability, p, never becomes greater than -^q/t
approaches 1 or certainty as a limit.
2) By choosing the quantity, t, which is less than Urn '^s, suffi-
ciently great, ice may bring ypq/t below any previously assigned
quantity, 8, or make the difference between p and ajs as small as we
please.
From these conclusions we obtain a fortiori the follow^lng
,. a
lim- = p.
This constitutes the essential features of the Bernoullian Theorem.
70. Poisson's Scheme. — Let pi denote the postulated prob-
ability for success in the first trial, p2. in the second, ps in the
112 LAW OF LARGE NUMBERS. [70
third, etc., and let furthermore qi, q-i, q^, • • • be the respective
probabihties for the correspoiuhng faihires. If the trial (observa-
tion) is repeated .s- times we obtain the following values for the
probable or expected value of the frequency for successes eia)
and the mean error e
e{a) = pi-\r p2-\- pz-\- •••Vs = ^Ph
€ = Vpir/i + p^q-i + Psqz + • • • Psqs = ^^p.qi {i = 1, 2, 3, • • -5)
If by po and qo we denote the arithmetic mean or the average
value of the s p's and s g's, such that
2^1 + p2 + P3+ ■■• Ps .^.
Po = -^ (3)
qo = , (4)
and assume that po and 70 denote the constant probabilities
during each of the s trials (observations), we should according
to the Bernoullian Theorem have :
eiois) = spo (5)
e{aj}) = yJspoqo (6)
where as stands for the absolute frequency in a Bernoullian
series.
An actual comparison of (1) and (5) and (3) shows that:
i'iap) = eias) (7)
where ap is the symbol for the absolute frequency in a Poisson
series. In other words: If the s trials had been ])erformed with
constant ])robability for success equal to ^>o instead of with
varying probabilities pi, p^, • • • /?«, the expected or probal)le
value would be the same for the Bernoullian and Poisson scheme.
With regard to the mean error we find, however, after a little
calculation,
^iKcc) = es\oi) - Zip, - poY. (8)
The expression for the mean error in Poisson's Theorem is of
the following form
€/- = Vpi^i + 2^292 + PzQz + • • 'PiQi = ^Zpiqi (i = 1, 2, 3- • -s).
70] poisson's scheme. 113
Now piqi may be transformed as follows:
Writing
Vi = Po -h (Pi — Vq)
Qi = qo - (Pi - Po)
and multiplying we obtain:
PiQi = PoQo - {pi - po)(po - qo) — {pi — PoY,
and summing up for all values of i from i = I to i = s we have:
€p' = spoqo — ^(Pi — Pq)' = f^B — '^{Pi — Po)^-
As {pi — 2Jo)" always is a positive quantity, it is readily seen that
the mean error in a Poisson scheme is always less than the mean
error in the corresponding Bernoullian series.
Writing e as follows:
€ = ^IplQl + 2?2<Z2 + • • • + Psqs
= J~. J P^ + • • • +P8 _ Pi^-\- • • • + Ps^
>( 5 S
and letting X = 'ss/t, we have according to Tchebycheff's The-
orem the following rule: The probability Pr that the relative
frequency remains inside the limits:
Pi + P2+ • '■ Ps ^ f ^l\ _ Pi -^ P2 + • • • -\- Ps
e
t
±
J)
1 JP I+ P 2-\- • • ' -\- Ps _ Pl^+P2^ -\- '" + Ps
t^ s s
is greater than 1 — (1/X") or 1 — (t'/s).
By taking t sufficiently large and by letting s approach infinity
as a limiting value the last term in the above difference, namely
the average probability, po, and X times the mean error, becomes
smaller than any previously assigned quantity, d, however small,
while Pt at the same time will approach 1 as a limit.
From this it now follows:
When an infinite number of trials is made on an event, following
the scheme of Poisson, then the expression:
,. a pi + P2 + • ■ ■ + Ps
hm - = = Pq.
x=tn S S
114 LAW OK LARGE NUMBEKS. [71
The essential part of Poisson's Theorem is contained in this
equation. ^Yhen p = p\ = jh = • • • pa y>'e have a BernoulHan
series and obtain:
hm- = p,
s-ao "^'
which result we already derived above in a direct way.
71. Relation between Empirical Frequency Ratios and Mathe-
matical Probabilities. — In the above limit, a indicates the total
number of lucky events while ,s' is the total number of trials, the
quotient a -^ s then is nothing- more than the empirical i)rob-
ability as defined in the precedinji; paragraphs. Both the
Bernoullian and Poisson Theorems show that this empirical
probability approaches the postulated a priori probability, p,
(or the average probability po) as a limiting value.
In this way we have succeeded in extending the theory of
probability to other ])r()blems than the conventional kind involved
in the games of chance or drawings of balls from urns. We do
not need to limit our investigations to problems where we are
able to determine a priori the probability for the happening of
an event in a single trial, but limit ourselves to postulate the
existence of such an a ])ri()ri probability.
A large number of trials or observations is made on a certain
event E. This event is now observed to have occurred a times
during the s total trials. To illustrate: An urn contains red
and white balls, the total number of balls being unknown, a
single ball is drawn and its color noted. This ball is replaced
and the contents of the urn is mixed. A second drawing is
made and the color of the drawn ball noted before the ball is put
back in the urn. Let this process be repeated .s- times, where s
is a large number, and furthermore let a be the number of red
balls which appeared during the s trials.
The quotient a -^ 5 we now call the empirical or a posteriori
probability for the observed event, in this particular case the
a posteriori probability for the drawing of a red ball. When
s = 00 the Bernoullian Theorem tells us that the empirical
probability found in this manner and the postulated a priori
probability whose niinicrical value, however, was unknown
before the drawings took j)lace, are identical as far as numerical
72] APPLICATION OF THE TCHEBYCHEFFIAN CRITERION. 115
magnitude is concerned. As we already observed in the intro-
ductory remarks to this chapter it is impossible to perform a
certain experiment an infinite innnber of times, and it is therefore
out of the question to determine the limiting and ideal value of
the posteriori probability, and we must satisfy ourselves with an
approximation by performing a finite number of trials, or let s
be a finite number. The quotient a -^ 5 is then the empirical
approximate a posteriori probability. We know also that al-
though this quotient is an a])i)roxiniation of the postulated a
priori ])robability only, that ))y increasing .? or what amounts
to the same thing, by making a large number of trials, the dif-
ference between the approximate empirical probability ratio,
a -T- s, and the a priori probability, p, becomes smaller as the
number of trials is increased. But how small is the difference?
Or how many times shall we repeat the trials (observations) so
that, for practical purposes, we may disregard this difference?
It does not suffice to be satisfied with the fact that the difference
becomes proportionately smaller the greater we make the number
of trials and merely insist that in order to avoid large errors it is
only necessary to operate with very large numbers. Immediately
the question arises: What constitutes a large number? Is 100
a large number, or is 1,000, 10,000, 100,000 or even a million an
answer to this question? As long as this question remains
unanswered, it ]i(>lps but little to j)()ke upon the "law of large
numbers," a tendency which unfortunately is too manifest in
many statistical researches by amateur statisticians. As long
as a definition, much less than a numerical determination of the
range of "small numbers" is lacking, little stress ought to be
laid on such remarks based in the metaphorical terms of "small"
and "large" numbers.
72. Application of the Tchebycheflfian Criterion. — It is readily
seen that even a rough quantitive determination of the difference
between the approximate a posteriori probability and the
postulated a priori probability based upon the mere vague state-
ment of "large numbers" is utterly impossible, and it remains
to be seen, therefore, if the theory of probability offers us a
criterion that might serve as a preliminary test for the above
difference. To restate our problem: If p is the postulated a 'priori
116 LAW OF LARGE NUMBERS. [72
probability and a -^ s is the empirical probability (a posteriori) or
relative frequency of the event, E, ivhat is the probobility that the
difference, \ (a's) — p | does not exceed a previously assigned quantity?
In the mean error and the associated theorem of Tchebycheff
we have a simple and easily applied criterion to test this prob-
ability.
Tchebycheff's rule states that the probability, Pt, of a devia-
tion of a variable from its probable value, not larger than X
times its mean error, is greater than 1 — (1/X").
For
X = 3 Pt> \ - I = 0.888
X = 4 7V > 1 - Jg = 0.937
X = 5 Pt> I - h = ^^-96.
This shows that a deviation from the expected or probable
value of the variable equal to 4 or 5 times the mean error possesses
a very small probability and such deviations are extremely rare.
Let us for example assume that the observed rate of mortality
in a certain population group is equal to .0200. Let furthermore
the number exposed to risk equal 10,000. The mean error is
(.02X.98\^
1 n onn ) ~ -0014. If the number of lives exposed to risk
was one million instead of 10,000, the mean error would be
(.02 X .98\^
"r^TTTTTT-TTT I =.00014. A deviation four times this latter quantity
1,000,000/ ^ ^
is equal to .00056, and according to Tchebycheff's criterion the
probability for the non-occurrcnre of a deviation above .00056
is greater than .937, or the probability of dying inside a year will
not be higher than .0206 or less than .0194. For an observation
series of 4,000,000 homogeneous elements we might by a similar
procedure expect to find a rate of mortality' between 0.02 +
0.00028 or 0.02 - 0.00028. Thus we notice that the mean error
of the relative frequency numbers decreases as the number of
observations increases.
CHAPTER X.
THE THEORY OF DISPERSION AND THE CRITERIA OF LEXIS
AND CHARLIER.
73. BemouUian, Poisson and Lexis Series. — In the previous
chapter we Hmited our disevission to single sets consisting of
s individual trials and found in the mean error and the criterion
of Tchebycheff a measure for the uncertainty with which the
relative frequency ratio a/s as well as the absolute frequency
a were affected. How will matters now turn out if, instead of a
single set, we make N sets of trials? As already mentioned in
paragraph 54, in general in N such sets we shall obtain A^ dif-
ferent values of a, denoting the absolute frequency of the event
represented by the sequence
ai, ao, 0(3, • • • oi^;.
Our object is now to investigate whether the distribution of
the above values of a around a certain norm is subject to some
simple mathematical law and if possible to find a measure for
such distributions.
In this connection it is of great importance whether the pos-
tulated a priori probabilities remain constant or not during the
N sample sets. Three cases are of special importance to us.^
1. The probability of the hapi)ening of the event remains
constant during all the N sets. The series as given by the ab-
solute frequencies in each set is knoAvn as a BemouUian Series.
2. The same probability varies from trial to trial inside each
of N sample sets, the variations being the same from set to set.
The series as given by the absolute frequencies is in this case
known as a Poisson Series.
3. Tlie probability remains constant in any one particular set
but varies from set to set. The absolute frequency series as
produced in this way is called a Lexis Series.
The above definition of these three series may, perhaps, be
made clearer by a concrete urn scheme.
1 The terminology in due to Charlier.
117
118 THE THEORY OF DISPERSION. [73
A. BernovlUan Series. — .9 balls are drawn one at a time from an
urn, containing black and white balls in constant proportion during
all drawings. Such drawings constitute a sample set. Let us in
this particular set have obtained say a\ white and /3i black balls,
where (X\-\- ^\ = -v. We make N sets of drawings under the
same conditions, keeping a record of white balls drawn in each
set. The number sequence thus obtained,
«i, oi-i, «3, • • • CX-N-
is a Bernoullian Series.
B. Poisson Series. — .9 individual urns contain white and
black balls, the proportion of white to black varying from urn
to urn. A single ball is drawn from each urn and its color noted.
In this way we get cvi white and /3i black balls constituting a set.
The balls thus drawn are replaced in their respective urns and a
second set of s drawings is performed as before, resulting in 02
white and jSz black balls. The number sequence,
Oil, "2, as, • • • a,v,
of white balls in N sets represents a Poisson Series.
C. Lexis Series. — s balls are drawn one at a time under the
same conditions as set No. 1 in the Bernoullian series. The ai
white and /3i black thus drawn constitute the first set. In the
second and following set the composition of the urn is changed
from set to set. The number sequence representing the number
of white balls in the N respective sets:
«!, ao, OC3, • • • Qfy
is a Lexian Series. The scheme of drawings is the same as in
the Bernoullian Series except that the proportion of white to
black balls varies from set to set.
74. The Mean and Dispersion. — Since we have no a priori
reasons for choosing any one particular value of the various as
of the above sequences in preference to any other, we might give
equal weight to each set and take the arithmetic mean as defined
by the formula :
\f — '^^'i + 0:2 + 0:3 + ■ • • q:.v ,j.
N ■ ^ ^
of the .V values of a.
73] BERNOULLIAN, POISSON AND LEXIS SERIES. 119
It will 1)0 unnecessary to enter into a detailed discussion of
the mean, which is a quantity used on numerous occasions in
every day life. We shall, however, define another important
function known as the dispersion (standard deviation). The
dispersion is denoted by the Greek letter, a, and is defined by
the formula
c' = ^ . (II)
We shall now attempt to find the expected value of the mean
and the dispersion in the three series. First of all take the
Bernoullian Series, Let the constant probability for success in
a single trial be j^o- We have then for the various expected values
or mathematical expectations of a:
Set No. 1: ^(0:1) = spo
Set No. 2: eia^) = spo
Set Nc N: e(av) = ^po
or:
e(ai) + ejaj) + • • • + e{a^) _ 7:e(aJ) _ Nspo _
N ~ y ~ N ~ "'^°'
which shows that the mean in a Bernoullian Series of N sample
sets is equal to the expected value of the absolute frequency in
a single set.
In regard to the dispersion we have for the various sets;
Set No. 1 : e(ai — My = e-{ai) = spoqo
Set No. 2: e{a2 — My — €-(0:2) = *Po9o
Set No. .V: eia^, - My = e\a^) = spoQo
Summing up and forming the mean we obtain for the expected
value of the dispersion in a Bernoullian Series, which we shall
denote by the symbol o-^ :
2€-(a,) Nspoqo
^B - ~]v~" ^ N "" *^°^°*
120 THE THEORY OF DISPERSION. [73
This result shows that the dispersion in a BernoulUan Series is
equal to the mean error, e, in a single set.
We now proceed to the Poisson Series. Let pi be the mathe-
matical probability of the happening of the event in the first
trial, p-2 be the pr()l)ability in the second trial and so on for all
trials, and let us furthermore denote the means of the p's and
g's by:
Pi + p2 -\- P3 • ■ ■ + ps
Po =
9o
s
qi + q2 + qs • " + Qs
Applying a similar analysis as above we have:
Set No. 1 : e(ai) = pi + p2 + • • • + Ps = spo
Set No. 2: eia^) = pi + P2 + •'■-{- Ps = spo
Set No. N: e(a^^) = pi -\- P2 + • • • + /?« = spo
The actual summation of the above values of e(a) gives us the
following value of the mean in a Poisson Series:
Mp = spo.
Let us for a moment assume that all the drawings had been
performed with a constant probability, po. According to the
Bernoullian scheme we should then have:
Ms = spo.
An actual comparison shows that M^ = Mp. This shows that
the same mean result is obtained if we draw .v balls from the urns
Ui, JJi, • • • Us with their corresponding probabilities pi, po, • • • p,
for drawing a white ball, as would be obtained if we drew all the s
balls from a single urn where the composition is such that the
ratio of the number of white to that of black balls is as po : qo,
where po and r/o are defined as above.
Let us now see how matters turn out in regard to the dispersion.
We have for the N sets:
Set No. 1 : e{ai - M)- = piqi + ^2^2 + • • • = ^p^q^ = e^ai)
Set No. 2: e(a2 — My = p^qi + ^272 + • • • = ^p^q^ — e^ia^)
Set No. N: e{a^ - Mf = p^q^ + ^292 + • • • = ^p.q. = t\a^)
73] BERNOULLIAN, POISSON AND LEXIS SERIES. 121
111 § 70 we showed, however, that ^p^Qv could be expressed
as follows:
€p-(a) = 52Jo'7o - 2(p^ - 2?o)- = 6/(a) - 2(p^ - po)^.
A simple straightforward calculation gives us now for the
dispersion, ap,
(t;~ = (T^ - l{p^ - /Jo)-,
In the corresponding Bernoullian Series with constant proba-
bility, ])o, the dispersion is equal to spo({o> which shows that the
dispersion in a Poisson Series is less than the corresponding
dispersion of the Bernoullian Series.
We finally come to the mean and the dispersion in the Lexian
Series which we shall denote by il/^, and cr^ respectively. Let us
furthermore define the two quantities po and Qq as follows:
Vi + P2+ •■■ +Pn
Vo =
qo =
N
9i + 92 + \- Qn
N
A computation along similar lines as above gives us first for
the mean, Mj^:
Set No. 1 : e(ai) = spi
Set No. 2: e(a2) = sp2
Set Xo. X: ^(«A-) = spj^
Thus we have:
Ic(q;..) Scvp, s[pi + P2+ ■•• p.vl
^^h = —^ = 7 Y = ]^ = m-
For the dispersion we have the following expectations:
Set Xo. 1: e(spo — aiY
Set Xo. 2: e(spo — a-iY
Set Xo. X: e{spo — a^Y
The expected value in the I'th set is
e{spo — aS~ = 2(*2?o — a^y<p^{a),
122 THE THEORY OF DISPERSION. [ 74
where ifXo^) is the general term in the i>robabiHty binomial:
{Pv + (IvY = 1- All analysis along similar lines as in § 65
gives us now:
e{spQ — aj- = s-pQ' — 2s-poPy + s-pj^ + sp,q^
= sp,.q,. + .v-(/J, - Pq)-
as the expected value of the square of the difference between the
mean and the absolute frequency in the j'th set. For all N
sets we then have
., ^sp„q,. .<?%
We have, however, the following identity:
2p^<7, = .Vpo^o - 2(p, — po)^
and hence
<^L = <^B + y— 2(/;, - PqY.
74a. Mean or Average Deviation. — Of quite another character
than the standard deviation or dispersion is the so-called mean
or average deviation, d, defined by means of the following
relation :
«i - M ! + I a2 - M I + I «3 - 3/ I + • • • + 1 «A— 3/ !
t? =
.V
where | a^ — M \ means the absolute difference between m, and
M. We shall now proceed to determine the expected value of
t? on the assumption that the observed data follow the Bernoullian
Ivaw. The mean in a Bernoullian series with constant prob-
ability po we found before to be equal to spo which was the
expected value of a in a single sample set of .9 trials. The
expected value of the absolute difference in the i^th set is therefore:
e\(x^ — spol = 2 I a^ — spo : (pM,
where as usual (pjoc) is the binomial probability function.
The deviations from spo are partly positive and partly negative.
We proved, however, before that
e{a^ — spo) = '^{a^ — spo)(p,(a) = 0.
74] MEAN OR AVERAGE DEVIATION. 123
Hence it is readily seen that the algebraic sum of the positive
deviations cancel the algebraic sum of the corresponding nega-
tive deviations so that e\a^ — spo \ equals twice the sum of
the positive deviations. Positive deviations occur for values
of a greater than spo, i. e., for all values which a may assume
from s to spQ in the binomial expansion: (po + ?o)*.^ Hence we
have (omitting subscripts) :
e\(x — sp\ = 2^{a — sp) { j p*^'""
= 2 j Za (M pV"" - ^p£ (1) Vq"-
The second of these sums represents the following function of
p and q
f(p, q) = p'+ {[) p^'q + (2 ) F^Y +••• + (,*) P'"'?''-
By ])artial differentiation in respect to p and by following
multiplication by p we have:
-PO^'"
Hence we mav write:
+ sp['l\p
apgsq
sq' ^
e\a — sp\ = 2^p- spf
dp
Furthermore f{p, q) is a homogenous function in respect to p
and q of the sth order. We may then apply the following well
known Eulerian Theorem from the differential calculus: If
f{p, q) is homogenous and has continuous first partial derivatives
then
Using this relation we may write:
e\ a - sp\ =^ 2 \p ^- spf V = 2pq
^Spo is taken to the nearest integer.
dp dq
124 THE THEORY OF DISPERSION. [74
The partial derivatives of f{p, q) with respect to y and q are
of the form:
dp ^ ^^ ^^ ^ ^ 1-2-3 •.. (5g- 1) ^ ^
. s(s — 1) ' • • sp
^ 1-2-3 •'■ sq P ^
df „ . s(s - 1) ■ ■ ■ {sp + 1)
Hence we have:
df _df_ s(s-D^-^sp r ■'^ „
69 ar/- 1-2.3 ■••.(7 ^ ^ '{spl sq\^ "^
We proved, however, in § ()3 that the expression inside the
bracket may be written approximately as follows:
T = '
yl'lTspq
This gives us finally (again using the subscripts) :
, lor l-^Poqo
e\a^ — spo\ = 2spoqolm = y
as the expected value of the absolute deviation in the j/th sample
set. This same relation evidently holds true for any other of
the N sample sets, which finally gives us the following result for ??:
^4
d = y- ■ ^spoqo.
^ IT
The dispersion in a Bernoullian series we found before to be
of the form:
(Tb = ^spoqo.
Hence we have the following relation between the dispersion
and the mean deviation:
(Tb
= ^^^= 1.2533^.
75. The Lexian Ratio and the Charlier Coefficient of Dis-
turbancy. — The results given in the last few paragraphs may be
embodied under the following captions.
75] THE LEXIAN RATIO. 125
1. The mean in a Poisson and Lexis Series is the same as the
mean in a Bernuullian Series with constant prubability of po
in a single trial, where po is defined as abooe.
2. The dispersion in a Poisson Series is less than the correspond-
ing dispersion in a Bernoullian Series.
3. The dispersion in a Lexis Series is greater than the dispersion
in a Bernoullian Series.
The mean and the dispersion of the Bernoullian Series occupy
in this connection a central j)(>sition and may be used as a standard
of comparison with other series. This is the method adopted by
Lexis in investigating certain statistical series, and we shall re-
turn to it in the following chapter. Lexis determines first
in a direct manner the dispersion as defined by formula (II)
from the statistical data as given by the number sequence a.
This process is known as the direct process (by Lexis called a
physical process) and gives a certain dispersion, a. After this
the dispersion is computed by an indirect (coml)inatorial)
process under the assumption that the series follows the Ber-
noullian distribution. The ratio, a : (Tq, which Charlier calls
the Lexian Ratio and denotes by the symbol, L, ma\' now give
us an idea about the real nature of the statistical series as
represented by the number sequence.
When L — I, the series is by Lexis called a normal series.
When L > I, the series is called hypernormal.
When L < 1, the series is a suhnonnal series.
It is easily seen from the respective formulas that the Poisson
Series are subnormal series whereas the Lexian Series are hyper-
normal. The great majority of statistical series are — as we
shall have occasion to see in the following chapter — of a hyper-
normal kind and correspond thus to the Lexian Series.
In § 74 we found the dispersion in the Lexis series as
<^l' = <yB~ + (.^- - s)a;-,
where
CrT =
.V
The quantity, o-p, is the natural measure of the variations in
the chances from the mean or normal probability, 2^0- It i^
12G THE THEORY OF DISPERSION. [75
however, dependent on the absolute values of these chances, so
that if all chances are changed in the same proportion, Cp is
also changed in the same proportion. Another drawback which
influences the Lexian Ratio is the variations of the number s
in each sample set. In order to overcome this difficiilty Charlicr
divides the. above quantity <jp by po- Assuming that the vari-
ations in the individual probabilities within each set are of no
perceptible influence on the dispersion, we have from the Lexian
dispersion:
"> o 2
Neglecting s in comparison with ,s'- and renjembering that
Ms = spo, we have as an approximation:
rp Va^^-2
0-,
Po M
B
Charlier calls the quantity lOOp the coefficient of disUirhancy of
the statistical series. It is readily seen that the Charlier coef-
ficient is zero in normal series. For hypernormal series it is a
positive real quantity whereas for subnormal series p is imaginary.
CHAPTER XL
APPLICATION TO GAMES OF CHANCE AND STATISTICAL
PROBLEMS.
76. Correlate between Theory and Practice. — In the theo-
retical analysis just completed we treated the fundamental ele-
mentary functions in the theory of probabilities, the probability
function, the expected or probable value of a variable quantity,
the mean error, the dispersion and the coefficient of disturbancy.
The formulas thus derived were founded upon certain hypo-
thetical axioms, which formed the basis of a mathematical a
priori probability as defined by Laplace. As far as the purely
abstract mathematical analysis is concerned it matters but little
if the hypotheses are physically true or not, that is to say, if
they agree with physical facts in the universe as it is known to
us. A mathematical analysis may be made on the basis of
widely divergent hypotheses, a fact which is clearly shown in
the Euclidean and Non-Euclidean geometries. It is, however,
quite a different matter when we wish to apply our theory to
actual phenomena (physical observed events) as it is evident
that a correlation between hypothesis and actual facts follows
by no means a priori. It is, of course, true that the different
hypotheses in the tlieory of probabilities are derived to greater
or less extent from outside sense data. Such sense data, however,
give us only the effect and no clue whatsoever to the relation
between cause and efl'ect. In the application of our theory every
hypothesis — or rather the results derived from such hypothesis
— must be verified by actual experience. Before such a veri-
fication is made, we advise the reader to be sceptical and not
trust too much in the authority of others but follow the sound
advice of Chrystal : " In mathematics let no man over-persuade
you. Another man's authority is not your reason." We can so
much more encourage an attitude of scepticism in view of the
fact that even among the leading mathematicians of the present
time there exists no uniform opinion as to the truth of the
axioms underlying the theory of probabilities.
127
128 APPLICATION TO GAMES OF CHANCE. [76
77. Homograde and Heterograde Series. Technical Terms.
— Whene\er a coiniiion charactoristic or attribute of several
groups of observed individual objects or events allows a purely
quantitative determination, it may be made the subject of a
mathematical analysis and in such cases we are often able to
make excellent use of the theory of probabilities. Such quan-
titative measurements may be divided into various domains
of classification. Traces of such classification are found in almost
every treatise on mathematical statistics but a uniform system
nomenclature is unfortunately lacking among the various
statisticians and any one reading the modern literature on mathe-
matical statistics notices often various inconsistencies of the
different authors. Mr. G. Udny Yule in his excellent treatise
"Theory of Statistics" classifies the statistical series into "sta-
tistics of attributes" and "statistics of variables." Apart from
the fact that Mr. Yule's statistics of variables also is a statistics
of attributes — although of different grades — the author appar-
ently ignores the criterion of Lexis and the associated criterion
of Charlier. The German writers use the terms "stetige und
unstetige Kollektivgegenstand " (continuous and discontinuous
collective objects), which were originally introduced by Fechner.
Other writers, such as Johannsen of Denmark and Davenport
of America, use still other terms. After ha^■ing made a com-
parison of the various systems of classification I have in the
following decided to adhere to the system of Charlier wherein
the observed statistical series are classified as homograde and
heterograde.
If the individuals all possess the same character or attribute
in the same grade (intensity) — or if we disregard the different
grades of the attributes — such individuals are called homograde,
and the statistical series thus formed is a homograde series. If
on the other hand we take into consideration the different
varying grades of the attributes obserxcd or measured and form
the series accordingly we obtain a heterograde series. As examples
of homograde series we may mention the observed recorded
series of coin tossing, card drawings in reference to a specified
event, number of births or deaths in a population group, etc.
A coin when tossed will either show head or tail, a person will
77] HOMOGRADE AND HETEROGRADE SERIES. 129
either be dead or alive. There are no intermediate degrees as
for instance that of a half dead person. In all such series the
dividing line between the occurrence of the event (attribute) E
and the occurrence of the opposite event E is distinct and suggests
itself a priori and there is no doubt as to the classification of the
observed event.
The original record of observation of a homograde series — also
known as the yrimary list — is simply a record of the presence or
non-presence of a specified attribute of the individuals belonging
to the group under observation and is of the following form:
Primary List of Homograde Individuals.
Attribute.
Sjmbol for the Indiyidual
Present {E).
Non-present {E).
/i
1
u
1
h
1
u
1
u
1
In this scheme the individuals I\, h and /a possess the attribute
E while the individuals /o and I^ do not have this attribute.
In observing the presence of a specified attribute in a group of
individual objects we meet, however, frequently series of quite
another nature than the simple homograde series. When in-
vestigating the different measures of heights of persons inside a
certain population group no simple dichotomous (f. e., cutting
in two) division in two opposite and mutually exclusive groups
suggests itself a priori. It is of course true that we might divide
the total population under observation into two subsidiary groups
of tall individuals and short individuals. But the question then
immediately arises, What constitutes a short or a tall person?
The answer must necessarily be arbitrary. Persons above the
height of 170 cm. may be classed as tall while persons falling
short of such measure may be classed as short persons, and we
might in this way form a primary homograde table of the form
as given above. There is no logical reason, however, to choose
the quantity 170 cm. as the dividing line and comparatively
10
130 APPLICATION* TO GAMES OF CHANCE. [78
little value would result from such a classification. It is evident
that all persons belonging to groups of tails or shorts are not iden-
tical as to the particular attribute in question. The height is
merely a characteristic which varies with each individual and no
two in(livi(hials have matheinatically speaking the same height.
If we take into consideration the different grades of height among
the individuals and arrange the primary table accordingly we
obtain a heterograde series of observations. The general form
of the primary table of such series is:
Primary
List
OF
Heterograde
Individuals.
Symbol
for the IndiTidual
Grade of Attribute.
/l
Xi
I2
X2
Is
x»
7*
Xt
/«
Xs
In X„
Here the quantities .ri, .ro, • • • .Tn give the measures (in kilo-
gram, liter, meter, etc.) of the characteristic in question.^
As examples of heterograde series we may mention the lengths,
volumes or weights of animals, plants or inorganic objects;
astronomical observations as to the brightness of celestial objects;
meteorological records of rainfall. tein])erature or barometer
heights ; the frequency of deaths among policyholders as to
attained age in an assurance company; duration of sickness or
disablement, etc.
The investigation of heterograde series is a ])r()bleni of which
we shall treat later under the theory of errors or frcxjuency curves.
The homograde series may, however, be explained fully by means
of the Bernoullian, Poisson and Lexian Series as founded on the
mathematical theory of probabilities in tlic ])re^■i()us chapters.
78. Computation of the Mean and the Dispersion in Practice.
— It would be sui>erfluous to enter into a detailed denionstratioii
of the practical calculation of either the mean or the dispersion
' It is to be noted that in the homograde series the primary list is given by
abstract numbers while the heterograde series consists of concrete numbers.
78] COMPUTATION OF THE MEAN. 131
were it not for the fact that this calculation is performed with a
lot of unnecessary and useless labor by the untrained student and
even by many professional statisticians. By the ordinary school
method the number zero is chosen as the starting point and all
the variables are expressed in their absolute magnitudes, i. e.,
their distance from 0. In this way one often encounters mul-
tiplication and addition of large numbers. The Danish biologist
and statistician, W. Johannson, has illustrated the futility of this
method in the following example taken from his treatise " Forela^s-
ninger over Lieren om Arvelighed" (Copenhagen, 1905).^ Dr.
Petersen, the director of the Danish Biological Station, counted
the tail fin rays of 703 flounders (Pleuronedes) caught around the
neighborhood of the Skaw. The observations follow:
Number of rays: 47 48 49 50 51 52 53 54 55 56 57 5S 59 60. 61
No. of flounders: 5 2 13 23 5S 90 134 127 HI 74 37 10 4 2 1
The ordinary way of computing the mean would be as follows:
[5 X 47 + 2 X 48 + 13 X 49 + h 1 X 61] 4- 703,
where 703 is the total number of individuals under observation.
In Chapter X we gave the following formula for the mean:
^r _ mi + mo + VH + • • • + w,v ,^.
M - -^ . (1)
This formula may evidently be written as follows:
mi — Mo + rrii — Mo -\- 7113 — Mo + • • • + m^ — Mq
M =
N
(2)
2(m, - Mo)
+ Mo = -^^ ^ + Mo = h + 3/,
In this expression Mo, which Charlier calls the provisional mean,
is an arbitrarily chosen number. To show how the introduction
of this quantity actually shortens the calculation of the mean
we return to the above quoted series of observations of tai', fin
rays of flounders.
1 German edition "Elemente der exakten Erblichkeitslehre" (Jena, 1913),
page 11.
132 APPLICATION TO GAMES OF CHANCE. [78
NtTMBEu OF Rays (x) ix 703 Flounders According to Observations op
Dr. Petersen.
A^ = ZF{x) = 703, Mo = 53.
Frequency
X.
= ^X-r)
X —
Mo.
(x - 3fo)F{x).
47
5
-6
- 30
48
2
-5
- 10
49
13
-4
- 52
50
23
-3
- 69
61
58
-2
-116
52
96
-1
- 96
53
134
+0
+
54
127
+1
+ 127
55
111
+2
+222
56
74
+3
+222
57
37
+4
+ 148
58
16
+5
+ 80
59
4
+6
+ 24
60
2
+7
+ 14
61
1
+8
+ 8
Sum = S
703
-373
+845
We have now:
b = (845 - 373) ^ 703 = 0.67, M = Mo + 6 = 53.67.
The method is quite simple and needs hardly any explanation.
From a cursory examination of the material we notice tluit the
mean is situated in the neighborhood of the series consisting of
53 rays. We choose therefore the provisional mean, J/o; as 53.
We next form the algebraic differences of x — Mq. These dif-
ferences are then multii)lied by F{x). The algebraic sum of
these products divided by N = 2F(.t) gives us the value of 6,
which quantity added to Mo gives the value of the mean, M.
To show a slightly modified form of the method we take the
following observations of coal-mine accidents in Belgium, covering
the i)eriod 1901-1910, from "Annales des Mines de Belgique."
These data I have reduced to a stationary population group of
140,000 mine workers. In other words the quantity s as defined
in § 83 is eciual to 140,000.
78] COMPUTATION OF THE MEAN. 133
Number (m) of Persons Killed in Coal Mine Accidents in Belgium,
1901-1910.
s = 140,000, N = 10, Mo = 140.
Year.
m.
1901
164
1902
1.50
1903
100
1904
130
190.-)
127
1906
133
1907
144
1908
1.50
1909
1.33
1910
133
Sum = 2
Hence
m — Mo.
(771 - MoY-
+24
576
+ 10
100
+20
400
-10
100
-13
169
— 7
49
+ 4
16
+ 10
100
- 7
49
- 7
49
-44 +68 1608
b = (68 - 44) ^ 10 = 2.4, .1/ = 140 + 2.4 = 142.4.
In this example probably it would have been easier to have formed
the sum 1,m^ directly and then obtained the mean by division
by 10. The actual formation of the algebraic sums of m^, — Mq
however, greatly facilitates the calculation of the dispersion, a,
to which we now shall turn our attention.
The formula for the dispersion
^'=^'-'^=-^'(^=1,2,3, ....V) (3)
may evidently be written as follows:
, (im - MoY- + (1112 - MoY + • • • + (w^ - MoY
N
2(m, - MoY
(4)
where h as usual means M — Mo, Mq being the provisional mean.
For Belgian coal mine accidents we thus obtain from the above
data:
0-2 = (1608 -^ 10) - .5.76 = 155.04.
Where the number of observed individuals is very large an
arrangement as that given above for the Belgian statistics becomes
too bulky and it is therefore customary to group the observations
in classes as for instance in the example of Dr. Johannsen. The
dispersion is then computed according to the following elegant
134
APPLICATION TO GAMES OF CHANCE.
78
method due to Charlier from whose brochure "Grunddragen af
den matematiska Statistiken" ("Rudiments of ^Mathematical
Statistics") I take the following example:
Number of Boys (//() pkk 500 Children Born in 24 Provinces of Sweden
DURING Each Month in 1883 and 1890.
s = 500, N = 576, Mo = 257, w = 5.
Class.
Limits
Hi.
Num
ber.
Frequency
xF\x).
x«i?(i).
{X + l)=F(x),
200-204
—
11
1
- 11
+
121
100
205-209
—
10
210-214
—
9
215-219
—
8
1
- 8
+
64
49
220-224
—
7
2
- 14
+
98
72
225-229
—
6
5
- 30
+
180
125
231-234
—
5
13
- 65
+
325
208
235-239
-
4
18
- 72
+
288
162
240-244
—
3
47
-141
+
423
188
24.5-249
—
2
60
-120
+
240
60
250-254
—
1
81
- 81
+
81
25.5-2.59
108
108
260-264
+
1
91
+ 91
+
91
364
265-269
+
2
60
+ 120
+
240
540
270-274
+
3
44
+ 132
+
396
704
275-279
+
4
22
+ 88
+
352
550
280-284
+
5
16
+ 80
+
400
576
28.5-289
+
6
6
+ 36
+
216
294
290-294
+
7
29.5-299
+
8
300-304
+
9
1
+ 9
+
81
100
Sum = 2
576
+ 14
+3596
+4200
The class width interval in the above scheme was chosen as 5.
The observed frequencies are given in column 3. We thus find
that the greatest frequency of 108 falls in tlic class interval
255-259. Choosing this class interval as the origin wc designate
the other class intervals with their proj)er positi\e and negative
numbers as shown in column 2. The provisional mean, Mq,
is taken as the center of class 0, or Mo = 257. In this way the
class interval ic — 5 is taken as the unit.
The whole calculation is very simple. We first of all form the
product X X F(x). The sum of these products divided with
57G = N gives the distance — b — from the ])r()\isioiiaI mean to
the arithmetic mean, expressed in units of the class interval, w.
78] COMPUTATION OF THE MEAN. 135
We have thus:
b = w X 14 -^ 576 = + 0.0243U' = + 0.122,
or
M = 257+ b = 257.12.
The formula for the dispersion takes the form
■^^^fi^)_j,y
N
where b is expressed in units of the class interval. The table gives
us
2F(.r).r2 _ 359(5 ^j.
a- = u'-[359G -^ 57G - (0.024)-] = ^^3.242,
a = ic X 2.498 = 12.49.
Charlier now checks the results by means of the following relation:
-(.r + \rF{x) = Z.v-F{x) + 2^F{x) + ZFix).
For the above example we have:
2.r2F(.r) = + 3,596
22.rF(.r) = + 28
2F(.r) = + 576
Sum - + 4,200 = 2(.r + l)-F{x),
which proves the accuracy of the calculation.
The full elegance of the Charlier self checking scheme is shown
at a later stage under the calculation of the parameters of fre-
quency curves. In the meantime the student may test the ad-
vantage of the provisional mean by trying to compute the mean
and the dispersion by the conventional school method. A
direct computation by this method would in the last example
take about a whole day's labor.
Before we proceed to apply the formulas previously demon-
strated, we wish to call the attention of the reader to the following
important properties of the mean and the dispersion:
1. The algebraic sum of the de\'iations from the mean — /. e.,
2(w„ — M) — is zero. This follows immediately from formula
(2) of §78. We have:
2(m,_- Mo)
N
M = _A_i_ ^+ j/^ = 5 + Mo,
136 APPLICATION TO GAMES OF CHANCE. [79
where Mo, the provisional mean, is an arbitrarily chosen number
and b = 1(7??,. — Mq) -^ N. If Mq = M we have evidently
b = 0, which proves the statement.
2. The dispersion (standard deviation) is the least possible
root-mean-square deviation, i. e., the root-mean-square deviation
is a minimum, when the deviations are measured from the mean.
We have (see formula (4)):
„ 2(/», - MY- Z(m. - MoY ,2
a- = jr = ^r 6^
from which tlic proposition follows a fortiori.
79. Westergaard's Experiments. — The Danish statistician,
Harald Westergaard, in his " Statistikens Teori i Grundrids"
gives the following results of 10,000 observations divided into
100 equal sample sets of drawings of balls from a bag containing
an equal number of red and white balls (the ball was returned
to the bag after each drawing):
White: 33 34 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Frequency. 01 12223 3 4565 11 95 10 48
White: 55 5d 57 .58 59 60 (il 62 63
Frequency: 3 5 4 4 111.
The elements as resulting from Westergaard's drawings clearly
represent a BernouUian Series where the number of comparison
s is equal to 100. Arranging the data in classes — taking 3 as
the class interval — the computation of the mean and the dis-
persion is easily performed by means of the Charlier self checking
scheme.
[{NOULLIAN
Series.
Number of
White Balls
IN 100
Drawings
(Westergaard).
s = 100, N = 100,
Mo = 49, w =
= 3.
m.
X.
Fix).
xF(x).
xH'ix).
(X+1)2F(X).
33-35
-5
1
- 5
25
16
36-38
-4
39-41
-3
5
-15
45
20
42-44
-2
8
-16
32
8
4.5-47
-1
15
-15
15
48-50
25
25
51-53
+ 1
19
+ 19
19
76
54-56
+2
16
-f32
64
144
57-59
+3
8
+ 24:
72
128
60-62
+4
2
+ 8
32
50
63-65
-1-5
1
+ 5
25
36
Sum
100
(-51-t-88)
329
503
^Uj ciiarlier's experiments. 137
Control Check.
Xx'^Fix) = 329
2'ExF{x) = 74
2F(.r) = 100
Sum = 503 = 2(0,- + l)2/'(a:)
b = w(88 - 51) : 100 = /r X 0.37 = 1.11,
or M = 3/o+ b = 50.11,
a^ = u'2[329 : 100 - 6-]^ = ic'-{3.29 - 0.137) = 28.377,
or a = 5.33.
Giving due allowance for the respective mean errors of the mean
and the deviation we have finally :-
.1/ = 50.11 it 0..53G, a = 5.33 d= 0..378.
We shall now compare these values with the corresponding the-
oretical values of the Bernoullian series. The a priori probabil-
ities of drawing red and white are in this example p — q — \.
Hence we have as the theoretical values for the mean and the
dispersion :
M^ = 100 X i = 50, <Js = VlOO XhXh = 5.
A comparison between the observed and the theoretical ideal
values — taking into account the proper mean errors — shows a
very close agreement as far as the dispersion is concerned while
the difference in the mean is about \ of the mean error. A
computation of the Lexian Ratio and the Charlier Coefficient of
Disturbancy yields the following results:
L = 1.072; lOOp = 3.68.
Taking into account the proper mean errors due entirely to
t\ie fluctuation of sampUmj we find, however, that our theoretical
results and formulas of the previous chapters have been verified
in an absolutely satisfactory manner.
80. Charlier's Experiments. — In the above mentioned bro-
chure, "Grunddragen." Charlier gives the results of a long series
of card drawings illustrating the BernouUian, the Poisson and
the Lexian Series. As an example showing the frequency dis-
• h is expressed in units of u\
^ For mean errors of M and a see Addenda.
138 APPLICATION TO GAMES OF CHANCE. [80
tributioii in a Bernoullian Series Churlier made 10,001) individual
drawings (with replacements) from an ordinary wliist deck and
recorded the number of black and red cards drawn in this manner.
Arranging the drawings in sample sets of 10 individual drawings,
]M. Charlier gives the following table:
Bernoullian Series. Number (w) of Black Cards in Sample Sets of 10.
.s = 10, N = 1,000, Mo = 5, IV = 1.
m.
X.
F(x).
xF^x).
x'^F(x).
(x +i)mx).
-5
3
- 15
+
75
+ 48
1
-4
10
- 40
+
160
+ 90
2
-3
43
-129
+
387
+ 172
3
_2
116
-232
+
464
+ 110
4
-1
221
—221
+
221
5
247
+ 247
6
+ 1
202
+202
+
202
+ 808
7
+2
115
+230
+
460
+ 1,035
8
+3
34
+ 102
+
306
+ 544
9
+4
9
+ 36
+
144
+ 225
10
+5
Sum: 1,000 - 67 +2,419 +3,285
CoxTKOL Check.
i:^xF{x) = + 2,419
2SxF(x) = - 134
1:F{x) = + 1,000
Sum = + 3,285 = 2(.r + iyF(x)
From the above values we obtain:
6 = - G7 : 1,000 = - 0.G7; a- = 2,419 : 1,000 - // = 2.415.
Making due allowance for mean errors we have thus:
M = 5 - 0.067 = 4.933 ± 0.050; a = 1.554 ± 0.035.
For the theoretical mean and dispersion we obtain the following
values: (v = q = h)
Mj,= 5; 0-^= 1.581,
which gives the following values for the Lexian Ratio and the
Charlier coefficient :
L = .983, loop is imaginary.
These results would indicate a slightly subnormal series. Tak-
ing into account the fluctuations due to sampling and for which the
80] charlier's experiments. 139
mean error serves as a measure the results become normal and
serve again as a verification of the theory.
Poisson Series. — As an illustration of the frequency distribution
in a Poisson .Series Charlier made the following experiment:
From an ordinary whist deck was drawn a single card and the
color noted. Before the second drawing a spade was eliminated
from the deck and replaced by a heart from another deck of
cards, so that tlie deck then contained 12 spades, 13 clubs, 13
diamonds and 14 hearts; from this deck another card w^as drawn
and the color noted. Then another spade was eliminated and a
heart substituted. P>om this deck, containing 11 spades, 13
clubs, 13 diamonds and 15 hearts, a card was again drawn. The
drawings were in this manner continued until all the spades were
replaced by hearts. The same operation was ai)plied to the
clubs, which were replaced by diamonds. After 27 drawings
the deck contained only red cards. Altogether 100 sample sets
of 27 drawings were made with the following results:
Poisson Series. Number (w) of Black Cards i.v Sample Sets of 27.
s = 27, N = 100, Mo = 7, w = I.
m. X. F{z). xF[x). x'^F(x). (i + l)2F(z). Control Check.
18
24 +378
14 + 32
+100
22
68 +510
126
128
25
12 +5 1 +5 +25 36
13 +6 1 +6 +36 49
Sum: 100 +16 +378 510
The calculation of the mean and the dispersion with their
respective mean errors yields the following result:
h= + 0.16, M = 7.16 db 0.211,
a-"- = 3.78 - (0.16)- = 3.754, a = 1.937 d= 0.149.
The theoretical Poisson values according to the formulas of
§ 67 are:
Mp - 6.75, (7p = 2.111.
If we now take the arithmetic mean of the various proba-
3
-4
2
- 8
+ 32
4
-3
6
-18
+ 54
5
_2
14
-28
+ 56
6
-1
14
-14
+ 14
7
22
8
+ 1
17
+ 17
+ 17
9
+2
14
+28
+ 56
10
+3
8
+24
+ 72
11
+4
1
+ 4
+ 16
140 APPLICATION TO GAMES OF CHANCE. [80
bilities of drawing a black card wo find that p^ — j. If all the
tlrawin^s had been i)c'rformed with a constant probability we
should accorcHiiij; to the Bernoullian scheme have:
Mj, = 27 X I = G.75, cr^ = V27 + i X f = 2.25.
These results verify the formulas as obtained under the discussion
of the Poisson Series. (Mp = Mg, Cp < cr^.)
Lexian Series. — In testing the Lexian Series Charlier first
took 10 samples of 10 individual drawings in each sample from
an ordinary whist deck. The number of black cards thus
drawn was recorded. After this, 10 samj)les of the same mag-
nitude were taken from a deck containing 25 black and 27 red
cards; and then 10 samples from a deck with 24 black and 28 red
cards. Of the total 270 samples (until the deck contains only
red cards) Charlier gives the first 100 which gave the following
result:
Lexian Series. Number (m) of Black Cards in 10 Drawings.
s = 10, A^ = 100, Mo = 4.
m.
X.
nx).
xF{x)
x^'Fix).
(x+-l)^F{x).
Control Check.
1
-3
4
-12
+ 3G
+ 16
2
-2
9
-18
+ 36
+ 9
3
-1
19
-19
+ 19
+
4-
21
+ 21
5
+1
23
+23
+ 23
+ 92
6
+2
10
+20
+ 40
+ 90
+294
7
+3
12
+36
+ 108
+ 192
+ 76
8
+4
2
+ 8
+ 32
+ .50
+ 100
Sum: 100 +38 +294 +470 +470
The final computations (with mean errors) give:
b= + 0.38, M = 4.38 db 0.167,
tr2 = 294 : 100 - b'- = + 2.796, <t = + 1.072 ± 0.118.
The mean probability in all trials was:
po = 21.50 : 52 = 0.4,135, or M^ = spo = 4.135,
ap = ^IspoQa = 1.557.
A calculation of the mean and the dispersion according to the
formulas under the Lexian Series (see § 74) gives according to
Charlier:
i/^ = 4.135, (Tl = 1.643.
81] EXPERIMENTS BY BONYNGE AND FISHER. 141
This shows that the dispersion in a Lexian Series is greater than
the corresponding BernouUian dispersion. The Lexian Ratio:
L = cTi^ : (Tii has the value LOG. The series according to the
terminology of Lexis has a hypernormal dispersion, although
a very small one. Charlier in "Grunddragen" (§ 30) says that
when arranging the material in 27 samples, each saiupit con-
taining 100 single trials, the Lexian Ratio has the value iy=3.82,
indicating a greater hypernormal dispersion than in the smaller
samples.
81. Experiments by Bonynge and Fisher. — As an additional
verification of the BernouUian, Poisson and Lexian Series my
co-editor, ]\Ir. Bonynge, and myself have repeated the experi-
ments of Westergaard and Charlier in a slightly modified form.
BernouUian Series. — \\\ 20 sample sets, each set containing
500 individual drawings, from an ordinary whist deck, I counted
the number of diamonds drawn in each sample. My records gave
the following scheme:
Bernoullian Series. Number of Diamonds (m) in 20 Sample Sets op
500 Drawings.
s = 500, N = 20, Mo = 125.
m. m — Mo. (m — ^o)*.
123 - 2 4
143 + 18 324
124 - 1 1
133 +8 64
142 + 17 289
130 +5 25
117 - 8 64
122 - 3 9
132 +7 49
109 -16 256
130 +5 25
139 + 14 196
138 + 13 169
129 +4 16
136 + 11 121
121 - 4 16
135 + 10 100
124 - 1 1
135 + 10 100
116 - 9 81
Sum: -44 +122 1,910
The results with their respective mean errors are as follows:
M = 128.9 ± 2.01, <T = 8.962 d= 1.416
142 APPLICATION TO GAMES OF CHANCE. [81
The theoretical Bernoulhan mean and the dispersion have the
values :
M^ = 125, a^ = ^Jfq = V500 X i X f = 9.682,
where p = I denotes the a priori probability of drawing a
diamond.
Again I counted the number of aces (irrespective of color)
which appeared in 100 sample sets of 100 individual drawings
from the same deck of cards. The records arranged in classes
gave the following scheme:
Number of Aces (to) in 100 Sample Sets of 100 Individual Drawings.
s = 100,
N = 100,
Mo = 8,
w = 1.
m.
X.
nx).
xF{x).
x^F{x).
(x+lWix).
Control Check
2
-6
1
- 6
36
25
3
-5
8
-40
200
128
4
-4
8
-32
128
72
5
-3
7
-21
53
28
6
-2
9
-18
36
9
7
-1
21
-21
21
8
13
13
9
+ 1
15
+ 15
15
60
10
+2
3
+ 6
12
27
11
+3
9
+27
81
144
+811
12
+4
1
+ 4
16
25
-110
13
+5
2
+ 10
50
72
+ 100
14
+6
2
+ 12
72
98
15
+7
+
801
16
+8
+
17
+9
1
+ 9
81
100
Sum: 100 -55 811 801
b= - 55 : 100 = - 0.55,
M = Mo+ b = 7.45 ± 0.279 (with mean error),
<^ = w^ ^.-,, X 6- = / .8075,
2/'(x)
or
a = 2.794 ±0.198 (with mean error).
The theoretical Bernoullian values are:
Mj, = 100 X i\ = 7.09, a^ = VlOO X tV X if = 2.663.
A comparison between tlie empirical and the theoretical a priori
values exhibits a close correspondence.
81] EXPERIMENTS BY BONYNGE AND FISHER. 143
Poisson Series. — As an illustration of the Poisson Series
Mr. Bonynge made the following experiment. A sample set of
20 single drawings of balls from an urn (one ball being drawn at a
time) was made under the following conditions:
In drawing No. 1 the urn contained 20 white and 20 black balls.
" 2 "
u
<(
21
a
(I
19
" 3 "
(<
(<
22
(<
II
18
■ ( It II nrv (( <( li oq II u 1 (< a
Altogether Bonynge took 500 sample sets which arranged in
classes give the following scheme:
Poisson Series.
Number
OF Black Balls (m)
in 500 Sample Sets
Individual Drawings (Bonynge).
s =
20, N
= 500, Mo
= 5.
m.
X.
J^x).
xF{x).
x'^Fix).
(x + \yF(x).
-5
2
- 10
50
32
1
-4
9
- 36'
144
81
2
-3
35
- 105
315
140
3
-2
52
- 104
208
52
4
- 1
86
- 86
86
5
109
109
6
+ 1
85
+ 85
85
340
7
+ 2
69
+ 138
276
621
8
+ 3
30
+ 90
270
480
9
+ 4
16
+ 64
256
400
10
+ 5
6
+ 30
150
216
11
+ 6
1
+ 6
36
49
Sum: S= 500 + 72 1876 2520
Hence we have :
h = 0.144, M = 5.144, (t^ = 3.732, a = 1.932.
The theoretical Poisson values are:
Mp = 5.25, <Tp = 1.86 (see formulas, § 74).
The mean of the various probabilities of drawing a black ball is
Vo — f ^- According to the Bernoullian scheme we should then
have the following values for the mean and the dispersion:
M^ = 20 X U = 5.25, a^ = (20 X U X ||)^ = 1.968.
These values confirm the Poisson theorems (Mp = M^, dp < ag).
Lexian Series. — As additional illustration of the Lexian Series
144 APPLICATION TO GAMES OF CHANCE. [81
I took 20 sample sets, each set containing 500 drawings of a
single ball from an urn (with replacements). The contents of
the urn varied from set to set as follows:
Sample set No. 1 : 20 white and 20 black balls.
« '< 2 : 21 " " 19 "
" " " 3 : 22 " " 18 " "
" " " 20 : 39 " " 1 " "
In the 21st set all the black balls were eliminated and the urn
contained white balls only. This set, however, was not taken in
consideration in calculating the mean and the dispersion.
Lexian Series.
Number
(m)
OF Black B
alls in 20 1
Sample Si
Individual Drawings (Fisher).
s =
500,
AT = 20, .
Mo = 130.
No. of Set.
111.
(w — Mo).
(m - MoY
1
251
+ 121
14641
2
240
+ 116
13456
3
222
+
92
8464
4
216
+
86
7396
5
193
+
63
3969
6
176
+
46
2116
7
183
+
53
2809
8
173
+
43
1849
9
156
+
26
676
10
135
+
5
25
11
140
+
10
100
12
127
- 3
9
13
115
- 15
225
14
96
- 34
1156
15
78
- 52
2704
16
69
- 61
3721
17
55
- 75
5625
18
43
- 87
7569
19
29
- 101
10201
20
19
- Ill
12321
Sum:
S =
- 539 + 661
99012
b = (661 - 539) : 20 = 6.6, M = Mo-\-b = 136.0 ± 15.86.
(^2 = 99012: 20 -62 = 4913.4, a = 70.098d=11.09 (with mean errors).
The theoretical Lexian values are:
Ml = 131.25, ar. = 72.676 (see § 74).
81] EXPERIMENTS BY BONYNGE AND FISHER. 145
If the series represented a true Bernoullian Series, we should
have
Mb = 500 X U = 131.25, as = V500 X f J X M = 9.839.
These values confirm the Lexian Theorem (Ml — Mb, o-l><tb)-
A computation of the Charlier Coefficient of Disturbancy from
the observed values gives :
lOOp = 50.80
whereas the theoretical value is 55.38, showing a decidedly
hypernormal dispersion, a result which was to be expected since
the probabilities of drawing black varies from | to ^^ in the
various sets of samples.
All the above experiments show a completely satisfactory
verification of the various theorems of the previous chapters
and may perhaps serve as a vindication of the followers of
Laplace, who like him hold that an a priori foundation for
probability judgments is indispensable.
11
CHAPTER XII.
CONTINUATION OF THE APPLICATION OF THE THEORY OF
PROBABILITIES TO HOMOGRADE STATISTICAL SERIES.
82. General Remarks. — In this chapter it is our intention to
discuss the application of the theory of probabilities to homograde
statistical series with special reference to vital statistics. We
owe the reader an apology, however, inasmuch as in the former
paragraphs we have employed the term .sfafistics without defining
its meaning in a rigorous manner. A definition may perhaps
appear superfluous since statistics nowadays is almost a house-
hold word. The term unfortunately is often employed as a mere
phrase without any understanding of its real meaning. This
applies especially to that band of self-styled statisticians, mere
dilettanti, who, with an energy which undoubtedly could be
better employed otherwise, attempt to investigate and analyze
mass phenomena regardless of method and system. When
investigations are undertaken by such dilettanti the common
gibe that "statistics will prove anything" becomes, alas, only
too true and proves at least that "like other mathematical tools
they can be wielded effectively only by those who ha\e taken the
trouble to understand the way they work."^
By the science of statistics ice understand the recording and
subsequent quantitative analysis of observed mass phenomena.
By mathematical statistics (also called statistical methods) we
understand the quantitative determination and measurement of the
effect of a complex of causes acting on the object under investigation
as furnished by previously recorded observations as to certain attri-
butes among a collective body of individual objects.
Practical statistics — if such a name may be used — then simply
becomes the mechanical collection of statistical data, i. e., the
recording of the observed attributes of each individual. In no
way do we wish to underestimate the importance of this process
> See Nunn, "Exercises in Algebra" (London, 1914), pages 432-33.
146
83] STATISTICAL DATA AND MATHEMATICAL PROBABILITIES. 147
which is as important for the statistical analysis as is the gathering
of structural materials for the erection of a large building.
Mathematical statistics is thus the tool we must use in the final
analysis of the statistical data. It is a very effective and powerful
tool when used properly by the investigator. At the same time
it is not an automatic calculating machine in which we need only
put the material and read off the result on a dial. A person
without any knowledge whatsoe^Tr about the nature of loga-
rithms may in a few hours be taught how to use a logarithmic
table in practical computations, but it would be foolish to view
the formulas and criteria from probabilities when applied to
statistical data in the same light as a table of logarithms in cal-
culating work. Such formulas and criteria must be used with
caution and discretion and only by those who have taken the
trouble to make a thorough study of probabilities and master
their real meaning and their relation to mass i)henomena. If
put in the hands of mere amateurs the formulas become as
dangerous a toy as a razor to a child.
It is not our intention to give in this work a description of the
technique of the collection of the material, which depends to a
large extent on local social conditions and for which it is difficult
to give a set of fixed rules. In the following we sluill treat the
mathematical methods of statistics exclusively, and furthermore
make the theory of probabilities the basis of our investigations,
83. Analogy between Statistical Data and Mathematical
Probabilities. ^Let us for the moment imagine a closed commun-
ity with a stationary population from year to year and let us
denote the size of such a population by s. Let us furthermore
suppose we w^ere given a series of numbers:
mi, mo, mz, • • ■ m^,
denoting the number of children born in various years in this
community. The ratios
mi mo mz mu
s ' ,s ' s ' s
may then be looked upon as probabilities of a childbirth in
various years. As Charlier justly remarks, "such an identi-
fication of a statistical ratio with a mathematical probability is
148 HOMOGRADE STATISTICAL SERIES. [83
ax first sight a mere analogy which possibly may have very little
in common Anth the observed statistical phenomena, but a
closer scrutiny shows the great importance for statistics of such
a view." If such ratios could be regarded as mathematical
probabilities wherein the various ??i's were identical to favorable
cases in s total trials, the mean and the dispersion could be de-
termined a priori from the Bernoullian Theorem. The founders
of mathematical statistics regarded the identification of an or-
dinary statistical series with a Bernoullian Series almost as
axiomatic. This view is found even among some leading writers
of the present time. Among others we apparently find this
traditional view by the eminent English actuary, G. King, in his
classic "Text Book." In Chapter II of this well-known standard
actuarial treatise a probability is defined as follows: "If an event
may happen in a ways and fail in (3 ways, all these ways being
equally likely, the probability of the happening of the event is
a -T- (a + /3)." With this definition as a basis King then de-
duces the elementary formulas of the addition and multiplication
theorems. He then continues: "Passing now to the mortality
table, if there be h persons living at age .v, and if these h+n survive
to age X -\- n, then the probability that a life aged .r will survive
n years is Z^+n ^ h = nPx- And again "tlie probability that a
life aged x and a life aged y will both survive n years is „PxX nVv"^
From the above it would appear that the author unreservedly
assumes a one-to-one correspondence between the U+n survivors
and "favorable ways" as known from ordinary games of chance
and a similar correspondence between the original J^ i)ersons and
"equally possible cases." A simple consideration will sliow that
there exists no a priori reason for such a uiiifiue correspondence
between ordinary empirical death rates and mathematical proba-
bilities. None of the original h persons can be considered as
' Mr. H. Moir in his "Primer of Insurance" tried to avoid the difTioulty by-
giving a wholly new definition of "equally likely events." According to
Moir "events may be said to be 'equally likely' when they recur with regu-
larity in the long run." Apart from the half metaphorical term "in the long
run" Mr. Moir fails to state what he means by the expression "with regu-
larity." If the statement is to be understood as regular repet it ions of a certain
event in various sample sets, it is evident that we may obtain a regular recur-
rence of the observed absolute frequencies in a Poisson Series, where — as
we know — the events are not equally likely." — A.F.
84] COMPARISON AND PROPORTIONAL FACTORS. 149
being "equally likely" as in the sense of games of chance.
Numerous factors such as heredity, environment, climatic and
economic conditions, etc., play here a vital part in the various
complexes embracing the original Ix persons.
The belief in an absolute identity of mathematical probabilities
and statistical frequency ratios seems to have originated from
Gauss. The great German mathematician — or rather the
dogmatic faith in his authority as a mathematician — proved
thus for a number of years a veritable stumbling block to a
fruitful devel()})inent of mathematical statistics. Gauss and his
followers maintained that all statistical mass phenomena could
be made to conform with the law of errors as exhibited by the
so-called Gaussian Normal Error Curve. If certain statistical
series exhibited discrepancies they claimed that such deviations
arose from the limited number of observations. The deviations
would become less marked if the number of observed values was
enlarged and would eventually disappear as the number of ob-
servations approached infinity as its ultimate value. The Gaus-
sian dogma held sway despite the fact that the Danish actuary,
Oppermann, and the French mathematicians, Binemaye and
Cournot, have pointed out that several statistical series, despite
all efforts to the contrary offered a persistent defiance to the
Gaussian law. The first real attack on tlie dogma laid down so
authoritatively by Gauss was delivered by the French actuary,
Dormay, in certain investigations relating to the French census.
It was, however, first after the appearance of the already men-
tioned brochure by Lexis, "Die ]\Iassenerscheinungen, etc.," that
a correct idea was gained about the real nature of statistical
series.
The Lexian theory was expounded in the previous chapters of
this work, and we are therefore ready to enter upon the investi-
gations of a few selected mass observations from the domain of
vital statistics.
84. Number of Comparison and Proportional Factors. — In
the mathematical treatment of the Lexian theory of dispersion
we tacitly assumed that the total number of individual trials in
a sample set or the number of comparison, s, remained constant
from set to set. In the observations on games of chance it
150
HX)MOGRADE STATISTILAL SERIES.
84
remained in our power to arrange the actual experiments in such
a manner that s would be constant. In actual social statistical
series such simple conditions do not exist. In comparing the
number of births in a country with the total population it is
readily noticed that the poi)ulation does not remain constant
but varies from year to year. For this reason the various
numbers m denoting the births are not directly comparable with
another. We may, however, easily form a new series of the form:
Wa',
s s s s
— • Vli, — • mo, — ■ 7??3, • ■ • —
*1 ^2 *3 -^A'
wherein the various numbers, mi, mo, ms • • •, corresponding to
the numbers of comparison Si, S2, s^, • • ■ , are reduced to a constant
number of comparison s. This series is by Charlier called a
reduced statistical series. Such a reduction requires, in many
Proportional Factoks for a Hypothetical Stationary Population in
Sweden and Denmark Equal to 5,000,000 and 2,500,000
Respectively.
Sweden,
npnmark,
Year.
Inhabitants.
a:ai,.
Year.
Inhabitants.
s:sk
1876
4,429,713
1.1288
1888
2,143,000
1.1666
1877
4,484,542
1.1150
89
2,161,000
1.1569
1878
4,531,863
1.1033
1890
2,179.000
1.1473
79
4,578,901
1.0919
91
2,195.000
1.1390
1880
4,565,668
1.0952
92
2,210,000
1.1312
81
4,572,245
1.0936
93
2,226,000
1.1230
82
4,579,115
1.0919
94
2,248,000
1.1121
83
4,603,595
1.0861
1895
. 2,276.000
1.0984
84
4,644,448
1.0765
96
2,3()6,(K)0
1.0841
1885
4,682,769
1.0677
97
2,33S,(H)0
1.0694
86
4,717,189
1.0600
98
2,371,000
1.0544
87
4,734,901
1.0560
99
2,403,000
1.0404
88
4,748,257
1.0.530
1900
2,432,000
1.0280
89
4,774,409
1.0472
01
2,4()2,00()
1.01.54
1890
4,784,981
1.0449
02
2,491,000
1.0036
91
4,802,751
1.0410
03
2,519.<)()()
0.9925
92
4,806,865
1 .0402
04
2. .546,000
0.9819
93
4,824,150
1 .0365
1905
2,574,000
0.9713
94
4,873,183
1.0261
06
2,603,000
0.9604
1895
4,919,260
1.0165
07
2.()35.()00
0.9488
96
4,962,568
1 .0076
OS
2,(')))S,()()0
0.9370
97
5,00n,()32
0.0981
09
2,7()2,()()0
0.92.52
98
5,062,918
0.9875
1910
2,7:',7,()00
0.91.34
1899
5,097.402
0.9809
11
2,80(),()()0
0.8929
1900
5,136,441
0.9734
1912
2,830,000
0.8834
85] CHILD BIRTHS IN SWEDEN. 151
cases, a certain correction. However, wiien the general ratios
s -7- Sk {k = I, 2, 3 • ■ ■ N) are close to unity the reduced series
may be treated as a directly observed series. In most of the
following examples taken from Scandinavian statistical tabular
works the proportional factor s -r- Sk, is close to unity as shown in
the table below. For Sweden I have, following Charlier, assumed
a stationary population s = 5,000,000. The corresponding
Danish s I have taken as 2,500,000.
The above figures are taken from " Sveriges officielle statistik "
and "Statistisk Aarbog for Danmark " for 1913 (Precis de
Statistique, 1913).
85. Child Births in Sweden. — From Charlier's "Grunddragen"
I select the following example showing the number of children
born in Sweden in the period from 1881-1900 as reduced to a
stationary population of 5,000,000.
Number of Children born in Sweden as to Calendar Year (Charlier).
s = 5,000,000, X = 20, Mo = 140,000.
Year.
m.
TO — Mo.
(to - A/o)'.
1881
145,230
+5,230
27,352,900
82
146,630
+6,630
44,089,600
83
144,320
+4,320
18,662,400
84
149,.360
+9,360
87,609,600
1885
146,600
+6,600
43,560,000
86
148,270
+8,270
68,392,900
87
148,020
+8,020
64,320,400
88
143,680
+3,680
13,542,400
89
138,300
-1,700
2,890,000
1890
139,600
- 400
160,000
91
141,070
+ 1,070
1,144,900
92
134,830
-5,170
26,728,900
93
136,540
-3,460
11,971,600
94
134,840
.
-5,160
26,625,600
1895
136,820
-3,180
10,112,400
96
135,330
-4,670
21,808,900
97
132,750
-7,2.50
52,.562,500
98
134,820
-5,180
26,832,400
99
131,320
-8,680
75,342,400
1900
134,460
-5,540
30,691,600
Sum 2 =
= + 53,190 -
- 50,390
654,401,400
From which we obtain:
6 = (+ 53,190 - 50,.390) : 20 = 140
M = Mo+b= 140,140
152 HOMOGRADE STATISTICAL SERIES. [86
a- = 654,401,400 : 20 - b"- = 32,700,470, or a = 5,718.
The empirical probability of a birth (po) is
po = M :s = ().()2S()3, so that qo = I - po = 0.97197 and the
Beriioullian dispersion
o-fi = ^spo f/o = 369.0.
The actual observed dispersion (5,718) is thus much greater
than the Bernoullian. The birth series is considerably hyper-
normal. The Lexian ratio has the value
L = 5,718 : 369.0 = 15.50,
while the Charlier coefficient of disturbancy is:
lOOp = 4.07.
Both the values of L and p show that the birth series by no
means can be conij)ared with the ordinary games of chance but
is subject to outward ]>erturbing influences.
86. Child Births in Denmark. — The following example shows
the corresponding birth series for Dermiark in the 25-year period
from 1888-1912 as reduced to a stationary population of 2,500,000.
The computation of the various parameters follows:
b = (39,713 - 30,287) : 25 = + 377,
M = Mo+b= 73,377,
cr2 = 281,208,156 : 25 - 6^ = 11,106,197.2,
cr/ = s2)o qo = 71,223. (po = M : s = 0.0293508),
1 = ^ .^^= 12.5
lOOp = 100( V^2 _ ^^2) . ^/ = 4 52.
NuMUEK OF Children Born in Denmark as to Calendar Year.
5 = 2,500,000, A^ = 25, .1/n = 73,000.
Year. m. m — Mo. (m — Mo)^.
1888 78,659 + 5,659 32,024,281
89 77,956 + 4,956 24,561,936
1890 76,154 + 3,154 9,947,716
91 77,377 + 4,377 19,158,129
92 74,059 + 1,059 1,121,481
93 76,965 + 3,965 15,721,225
94 75,9.56 + 2,956 8,740,636
1895 75,649 + 2,649 7,017,201
96 76,183 + 3,183 10,131,489
97 74,404 + 1,404 1,971,216
86
CHILD BIRTHS IN DENMARK.
153
Year.
m.
m
■-Mo.
(m-A/o)2.
98
75,570
+
2,570
6,604,900
99
74,236
+
1,236
1,527,()06
1900
74,146
+
1,146
1,313,316
01
74,341
+
1,341
1,798,281
02
73,058
+
58
3,364
03
71,802
- 1,198
1,435,204
04
72,359
- 641
410,881
1905
70,981
- 2,019
4,076,361
06
71,280
- 1,720
2,958,400
07
70,516
- 2,484
6,170,256
08
71,438
- 1,567
2,455,489
09
79,597
- 2,403
5,774,409
1910
68,777
- 4,223
17,833,729
11
66,016
- 6,984
48,776,256
1912
85,952
- 7,048
49,674,304
Sum: 2
= -30,287
+39,713
281,208,156
Practically the same deductions hold true for this Danish
series as for the Swedish series. We meet again a hypernormal
series subject to perturbing influences. The closeness of the
two values of the Charlier coefficient of disturbancy indicates
that the number of births in Sweden and Denmark apparently
are subject to the same outward disturbing influences.
87. Danish Marriage Series. — The following table shows the
number of marriages in Denmark from 1888-1912.
Number of M
ARRIAGES ]
[N Denmark.
&■ = 2,500,000,
.V = 25, .
Mo = 18,000.
Year.
m.
m — Ma.
(m - ilfo)».
1888
17,605
- 395
156,025
89
17,622
- 378
142,884
1890
17,181
- 819
670,761
91
17,017
- 983
966,289
92
17,012
- 988
976,144
93
17,676
- 324
104,976
94
17,445
- 555
308,025
1895
17,736
- 264
69,696
96
18,239
+ 239
57,121
97
18,676
+ 676
456,976
98
18,870
+ 870
756,900
99
18,661
+ 661
436,921
1900
19,015
+ 1,015
1,030,225
01
17,870
- 130
10,900
02
17,712
- 288
82,944
03
17,791
- 209
43,681
04
17,895
- 105
11,025
1905
17,947
- 53
2,809
154 HOMOGRADE STATISTICAL SERIES. [87
Year.
m.
06
18,592
07
19,072
08
18,750
09
18,453
1910
18,255
11
17,749
1912
18,034
- 251
in—Mo.
(m-Afo)".
+ 592
350,464
+ 1,072
1,149,184
+ 750
562,500
+ 453
205,209
+ 255
65,025
63,001
+ 34
1,156
Sum: 2 = -5,742 +6,617 8,686,841
Hence we have:
b = (6,6-17 - 5,742) : 25 = 35, M = Mo + 6 = 18,035.
<T- = (8,686,841 : 25) - b^- = 346,249, a = 588.43,
(Ti, = 133.81, L = 4.41, lOOp = 5.73.
We encounter again a hypernormal series with quite large
perturbations. For Sweden Charlier has computed che coef-
ficient of disturbancy for marriages in the period 1876-1900 and
found it to be 5.49. A comparison with the same quantity for
the above Danish data shows that the perturbing influences
for the two countries are about the same.
88. Stillbirths. — As another example from vital statistics I
give the number of stillbirths in Denmark from 1888-1912 as
compared with a hypothetical number of 70,000 births per annum.
Number of Stillbirths in Denmark as Reduced to a Stationary Number
OF 70,000 Births per Annum.
s = 70,000, N = 25, Mo = 1,700.
Year. m. m— Mo. (m — Afo)».
1888 1,861 + 161 25,921
89 1,924 + 224 50,176
1890 1,830 + 130 16,900
91 1,779 + 79 6,241
92 1,811 +111 12,321
93 1,788 + 88 7,744
94 1,719 + 19 361
1895 1,7.53 + 53 2,809
96 1,714 + 14 196
97 1,811 + 111 12,321
98 1,797 + 97 9,409
99 1,7.37 + 37 1,369
1900 1,696 - 4 16
01 1,732 + 32 1,024
02 1,694 - 6 36
03 1,685 - 15 225
04 1,682 - 18 324
89] COAL MINE FATALITIES. 155
Year. m. M-mo. (m— Mo)*.
1905 1,705 +5 -25
06 1,620 - 80 6,400
07 1,723 + 23 529
08 1,694 - 6 36
09 1,665 - 35 1,225
1910 1,658 - 42 1,764
11 1,6.59 - 42 1,764
12 1,638 - 62 3,844
Sum: 2 = -310 +1,184 161,216
Actual computation gives:
h = (1,184 - 310) : 25 = 34.9G, .1/ = 1,734.96,
<j- = 101, 21() : 25 - h- = 5,220.44, lOOp = 3.407.
The series is again liypernonnal. We shall show presently,
when discussing the disturbing influences, that this series after
the elimination of the secular perturbations actually represents a
normal series. In the meantime we give a few examples relating
to accident statistics.
89. Coal Mine Fatalities. — The following table gives the
number of deaths from accidents in coal mines in various countries
in the period 1901-1910 together with the number of compari-
son s.
United
Year Belgium Austria England France Germany Japan States
s = 140,000 s = 68,000 s = 900.000 s = 180,000 8 = 500.000 s = 110,000 s = 610,000
1901 164 81 1,224 218 1,170 263 1,982
02 150 73 1,116 196 995 188 2,263
03 160 50 1,134 184 960 278 1,952
04 130 62 1,116 193 900 239 2,135
1905 127 99 1,215 187 930 3.54 2,214
06 133 70 1,161 1,262 985 .578 2,944
07 U4 73 1,179 198 1,240 399 2,977
08 150 58 1,188 171 1,355 262 2,220
09 133 73 1,287 210 1,021 667 2,440
1910 133 63 1,530 194 985 245 2,391
This gives the following values for the Charlier coefficient:
lOOp
Belgium 2.55
Austria 13.85
England 4.71
France 34.19
Germany 9.27
Japan 44.121
U. S. A 12.07
1 1 doubt whether the Japanese data as given by the Bureau of Mines are
reliable.
156 HOMOGRADE STATISTICAL SERIES. [89
The comparatively large values of p show that the fatal ac-
cidents in coal mines are subject to violent perturbations. The
disturbinj:!; influences are greatest for France where the Charlier
coefficient is above 3-i, which immediately shows that some
powerful disturbing influence has made itself felt. Looking over
the table we find a very large number of deaths for the year 1906.
The extremely heavy death rate in this year was caused by the
Courrieres mine explosion, in which 1,099 persons lost their lives
and marks j^robably the most fatal disaster in the whole history
of coal-mining. Eliminating this catastrophe from the data in
the table given above we find indeed that the coefficient of dis-
turbancy becomes imaginary, indicating very stable conditions
in French mines. Thus eliminating the more fatal catastrophes
we get at least for France a subnormal series for the everyday
accidents. In order better to illustrate the influence of the
elimination of the most disturbing catastrophes I submit the
following two series as reduced to a stationary s = 630,000 of
fatal coal mine accidents in the United States in the period
1900-1914 as recorded by the Bureau of Mines. The first series
shows total number of deaths nik, the second series gives the total
deaths nik' per year after eliminating all such accidents in w^hich
5 or more men were killed.
Number of Deaths from Accidents in Coal Mines in United States.
s =
■ 630,000, N =
= 15.
mic
Ttlt'
fj%ii
TO*'
1900
2,173
1,843
1908
2,293
1,967
01
2,048
1,863
09
2,520
2,053
02
2,337
1,837
1910
2,470
2,085
03
2,016
1,768
11
2,3.50
1,984
04
2,20.5
1,911
12
2,060
1,839
1905
2,286
1,964
13
2,3.50
1,957
06
2,111
2,075
1914
2.070
1,810
07
3,074
2,190
The first series gives a coefficient of disturbancy equal to 11.06
while the same cpiaiitity for the second series has the value 5.51.
Despite the fact that tlie coefficient of disturbancy is reduced
about 50 per cent, there still remains disturbing influences, which
clearly shows that conditions in American mines are not so stable
as in the mines of France, Belgium and England.
90] REDUCED AND WEIGHTED SERIES IN STATISTICS. 157
90. Reduced and Weighted Series in Statistics. — So far all
our problems in statistical analysis have been related to series
where the value of s was constant or where the ratio s : Sk was
so close to unity that it might be used as a factor of propor-
tionality. We shall now consider the case where this ratio differs
greatly from unity. As an illustration of this kind of series I
choose the number of fatal coal mine accidents in various states
of the American Federation together with the number of people
engaged in coal mining in these states. The figures as taken from
the report of the Bureau of Mines relate to the year of 1914.^
Number of Persons Engaged in Mining (sii) and Number Killed
(wii) in 20 States During the Year 1914.
s = 1000. .V = 20.
s*. mic. posk- Itoj— po«A:l-
1 Alabama 24,552 128 73 55
2 Colorado 10,550 75 31 44
3 Illinois 79,529 141 237 96
4 Indiana 22,110 44 66 22
5 Iowa 15,757 37 47 10
6 Kansas 12,500 33 37 4
7 Kentucky 26,332 61 79 18
8 Maryland 5,675 18 17 1
9 Missouri 10,418 19 31 12
10 New Mexico 4,021 18 12 6
11 Ohio 45,815 62 136 74
12 Oklahoma 8,948 31 27 24
13 Pennsylvania 175,745 595 524 71 (Anthracite Mines)
14 Pennsylvania 172,196 402 513 111 (Bituminous Mines)
15 Tennessee 9,580 26 29 3
16 Texas 4,900 11 15 4
17 Virginia 9,162 27 27
18 Washington 5,730 17 17
19 W. Virginia 74,786 371 223 148
20 Wyoming 8,353 51 25 26
Sum: 2 = 726,659 2,167 709
It will be noted that the population engaged in mining varies
greatly from state to state. In making a simple reduction to a
common number of comparisons by a proportional factor it is
evident, however, that we would give the same weight to the
observed from New INIexico with a population of miners equal to
1 Catastrophes in the Eccles Mine in \\'est Virginia and in the Royalton
Mine of Illinois are eliminated.
158 HOMOGRADE STATISTICAL SERIES. [90
4,021 as to the mining population of the state of Pennsylvania
where over 340,000 persons are engaged in the same industry.
This procedure is faulty. Let us imagine for the moment two sets
of drawings from a bag containing white and black balls. The
first sample set contained 10,000 drawings and the second set
only 100 drawings. If these series were reduced to a common
number of comparison s = 1,000 we should have
1,000 , 1,000 , , J. P ,
m 1 and ~T;^<r '"2 (wi and ???2 standmg tor the number
of white balls) as tlie number of white balls drawn in sample sets
of 1,000 single drawings.
But these values are not eciually rehable. The mean error in
the second series is in fact 10 times as large as the mean error in
the first series. In order to overcome this difficulty we ask the
reader to consider the following series:
The element — ?7?i is repeated si times
tt u
t( a
'-m, "
i<
St
— ms
cc
S2
S3
In this way we obtain a series with Si + So + ■'?3 + ■ • • + ^at
elements which may be termed a reduced and iceigJited series
since the larger Sk appears oftener than the smaller values of Sk-
We shall now see if it is possible to determine the expected value
of the mean and the dispersion if the series is supposed to follow
the Bernoullian Law.
The mean is defined by the following relation:
-mi
-sr
s s s
M= -miH H WiH 7/12+ •••+- W2
90] REDUCED AND WEIGHTED SERIES IN STATISTICS. 159
< Sy »
s s \
+ • • • + — m^v + • • • + — Wjv M- [*i + 52 + • • • + .?vl
Denoting the average empirical probability by po we have
2mfc : 25fc = po and,
J/^ = .s'Po.
As to the dispersion it takes on the following form:
''' = [w'''~'^''J'^ ---^ik"''''^''}
<: S Y >
-^ [^1 + ^2 + • • • Sy\
Hsk -TTik — spo ) 2 - {nik—SkPo)-
In finding the theoretical dispersion, assuming a Bernoullian
distribution for which po may be used an an approximation of
the mathematical a priori probability, we ask the reader to
examine the general term of the expression for a", viz.:
^ (mk — •npo)- : ^Sk.
If the individual trials follow the Bernoullian Law the expected
value of the factor {rtik — SkPoY takes the form:
e[(mk — .s'kpo)-] = 2(»^. — .n-poYtpinik) = Skpoqo.
This brings the general term for cr- to the form:
o
S' S
^ PoQo = ^r- spoqo-
160 HOMOGRADE STATISTICAL SERIES. [90
Thus the expected value of a- accordhig to the Bernoulliaii
distribution may be written as follows:
^•=-v g jV"*
o-b" = 2 s^spoqo = -^-spoqo, or : a^ = fVspoQo,
where as before /^ = -"U- : -^t find / = ^Z-
JNs_
These formulas give us the means of computing the Lexian
Ratio and the Charlier coefficient of disturbancy in the ordinary
way. Some of the computations require, however, a great
amount of arithmetical work and the goal is reached more
easily by making use of the mean deviation (in § 74a).
We found there the following relation:
In the w^eighted series it is readily seen that the value of d
will be of the form:
\Sk
Z^j vik — Skpo
^Sk ^Sk
If the series may be assumed to follow a Bernoullian distri-
bution we have
o-g = 1.253:]t?.
P'rom the above formulas it is readily noticed that we may find
the mean and the dispersion directly from the observed series
without a preliminary reduction to a common number of com-
parison s. This is in fact the method used in the above example
of coal-mine accidents in various states. We have:
2^0
Znik : ^Sk = 2,167 : 726,659 = .002982,
^ ^ 2.9! w, — Skpoj ^ ^^^^^ ^ ^^^ . 726.659 = 0.9757,
a = 1.253.3 X I? = 1.223,
"^0 000
(Ts' = P-nm^ = ;^.,^ X 1,000 x 0.997 x 0.003 = 0.O817,
lOOV^-cr^^
lOOp = ^ = 40 approx.
91 ] SECULAR AND PERIODICAL FLUCTUATIONS. 161
The large value of the Charlier coefficient of disturbancy
clearly shows that conditions in coal mines by no means are
uniform in the whole union but vary greatly according to the
locality. An actual computation shows in fact that in a few
states such as ^Michigan and Iowa we find an imaginary coeffi-
cient of disturbancy whereas States as Ohio and West Virginia
exhibit marked hypernormal series with a large coefficient of
disturbancy. The establishment of this fact is of some im-
portance in connection with accident assurance. Many sta-
tisticians seem to be of the opinion that a standard accident table
computed from the data of the whole union ought to serve as
the basis for assurance premiums. Such a table would assume
uniform conditions all over the union. The enormously high
value of p as computed above shows the fallacy of such a view.
91. Secular and Periodical Fluctuations. — In the last para-
graphs we have just learned how to detect the presence of dis-
turbing influences in a statistical series. A value of the Lexian
ratio differing from unity or a value of the Charlier coefficient of
disturbancy differing from zero indicates the presence of fluc-
tuations in the chances for the event or phenomena under in-
vestigation. After having established the presence of such
fluctuations it is the duty of the statistician to trace the sources
of the disturbing influences. This is in general done by means of
the theory of correlation, which will be discussed in the second
volume of this work.
It is, however, possible to classify the disturbances under two
categories which by Charlier are termed as secular and periodical
variations.^ The periodical fluctuations are in general difficult
to discuss on account of the variations in the period of the dis-
turbing forces. In many cases we are in absolute ignorance
about the length of such a period and therefore unable to subject
the series to a mathematical analysis. If the length of the period
is known it is indeed not difficult to determine the periodical
disturbances. This is often the case in series giving the occur-
rence of a certain disease in various months. In statistics giving
the frequency of malaria in a community the observed cases are
1 Lexis uses the terms "evolutionary" ("symptomatic") and "periodical"
("oscilating") for such fluctuations.
12
102 HOMOGRADE STATISTICAL SERIES. [91
nearly all limited to the warmer months and infrequent in the
winter months.
In the secular fluctuations due to certain outward influences
workinj; continually in the same direction it is quite easy to
calculate the rate of such variations.
Let /3 denote the increase (decrease) of the original probabilities
(pi, p-2, Pa, ■ ■ • Pn) from set to set in the given statistical series
so that
P2 — Pi = /3
Ps — P2 =
Pn - Pn-i = /3
We then have:
Pk = Pi + a- - i)/3. (1)
The mean probability has the value:
Pl + P2 + P3 + ■ ■ ■ + Pn
Po =-
N
_ Pi + Pi + ^ + Pi + 2/3 + • • • + Pi + (iV - 1)^ ,.,,
N ^"^
= Pi + ^^/?.
Eliminating pi from (1) and (2) we have:
Pk - Po= (^ ^- - 2 ) ^'
If the observed and reduced numbers »?i, Wo, '"3, • • • ^y may be
regarded as approximately coinciding with ,s7>i, sp^, sp^, • ■ ■ sp.^
we may write (2) as follows:
('■ - m
In order to obtain an expression for .<f/3 in known quantities we
must climate the quantity k. ]\Iulti])lying both sides of the
equation (3) by k — (X -\- l)/2 we have:
{rrik
-,„(/.--l±i) = (;.-^^)V
91 ] SECULAR AND PERIODICAL FLUCTUATIONS. 163
Summing this expression for all values k from k = 1 to k = N
we have:
.(._^),.,_,,)_.,,(,_^y
(4)
The following expressions from the summation of series are
well known to the reader from elementary algebra:
Z A- = ^ A^(A^ + 1),
Substituting these values in (4) we obtain after a few simple
trar^iormations the following expression for .s'/3:
12 / N + l\
Tr '.ECULAK Annual Decrease OF Number of Stillbirths in Denmark.
s = 70,000, .V = 25, M = 1,735
* 2~
Year.
k.
mk.
mt-
-M.
1SS8
1
1,SG1
+ 126
-12
89
2
1,924
+ 189
-11
1890
3
1,830
+ 105
-10
91
4
1,779
+
44
- 9
92
5
1,811
+
76
- 8
93
6
1,788
+
53
- 7
94
7
1,719
—
16
- 6
1895
8
1,753
+
18
- 5
96
9
1,714
—
21
- 4
97
10
1,811
+
76
- 3
98
11
1,797
+
62
- 2
99
12
1,737
+
2
- 1
1900
13
1,696
—
39
01
14
1,7:32
-
3
+ 1
02
15
1,694
—
41
+ 2
03
16
1,685
-
50
+ 3
04
17
1,682
—
53
+ 4
1905
IS
1,705
-
30
+ 5
06
19
1,602
_ 1
115
+ 6
07
20
1,723
—
12
+ 7
08
21
1,694
-
41
+ 8
09
22
1,665
—
70
+ 9
1910
23
1,658
-
77
+ 10
11
24
1,658
-
77
+ 11
1912
25
1,638
-
97
+ 12
(*~^2^)^'"*-
M).
- 1,512
- 2,079
- 1,C50
- 396
- 808
- 371
+ 96
- 90
+ 84
- 228
- 124
- 2
- 3
82
- 150
- 212
- 150
- 690
84
- 328
- 630
- 770
- 847
- 1,164
Sum: -11,590
1()4 HOMOGRADE STATISTICAL SERIES. [91
As ail exaraplo illustrating secular fluctuations I take the
previously discussed series of stillbirths in Denmark.
Wc luive in this case
hence;
s(3= - 11,590 : 1,300 = - 8.92.
From this we may draw the conclusion that the number of still-
births in Denmark pr. 70,000 births per annum on the average
is decreased by 8.92.
If the fluctuations are of an essential secular character we may
write
m = M + {k- 13)(- 8.92)
as the number of stillbirths pr. annum. Apart from accidental
fluctuations due to sampling we should therefore obtain a nearly
normal series for the 2r)-vear period if we calculated the number
of stillbirths each year according to the expression: iiik
— {k — 13)(— 8.92). Such a computation is given below:
Number of Stillbikths in Denmark Freed from Secular Fluctuations.
.s = 70,000, N = 25.
Year. k. to*- (it - 13) (- 8.92). Year. k. m*- (fc - 13)( - 8.92).
1888 1 1,754
89 2 1,826
1890 3 1,741
91 4 1,699
92 5 1,740
93 6 1,726
94 7 1,066
1895 8 1,708
96 9 1,678
97 10 1,784
98 11 1,779
99 12 1,728
A computation of the characteristics of this series gives:
M = 1,735, a = 37.09, <t,i = 41.0, lOOp imaginary.
The dispersion is now slightly subnormal and the coefficient
of disturbancy is imaginary whereas in the original series in
§ 88 it had a value equal to 3.4.
1900
13
1,696
01
14
1,741
02
15
1,712
03
16
1,712
04
17
1,718
1905
IS
1,730
06
19
1,675
07
20
1,875
08
21
1,765
09
22
1,745
1910
23
1,747
11
24
1,756
1012
25
1,745
92] CANCER STATISTICS. 165
92. Cancer Statistics. — Mr. F. L. Hoffman in his treatise " The
MortaUty from Cancer Throughout the World" gives some very
interesting statistics on mortaHty from cancer in various h)cahties.
Through the kin(hiess of ]Mr. Hoffman I am able to submit the
following series relating to cancer among males in the City of
New York (Manhattan and Bronx Boroughs) :
Deaths from Cancer {rrik) in the City of New York as Reduced to a
Stationary Population of 1,000,000.
Year.
1889
1890
91
92
93
94
1895
96
97
98
99
1900
01
02
03
04
1905
06
07
08
09
1910
11
12
1913
A computation of the dispersion and the Charlier coefficient
of disturbancy gives a value of lOOp in the neighborhood of 18,
indicating marked fluctuations. An inspection of the series shows
immediately that there is a marked increase in the rate of death
from cancer. Working out the secular disturbances in the ordi-
s = 1,000,000,
N = 25,
M = 560.
h.
rrik.
mii—M.
-T- ('-
^-^^)(..-M)
1
377
-183
-12
2,196
2
476
- 84
-11
924
3
410
-150
-10
1,500
4
444
-116
- 9
1,044
5
462
- 98
- 8
784
6
423
-137
- 7
959
7
442
-118
- 6
708
8
493
- 67
- 5
335
9
505
- 55
- 4
220
10
515
- 45
- 3
135
11
513
- 47
- 2
94
12
547
- 13
- 1
13
13
595
+ 35
14
540
- 20
+ 1
-20
15
580
+ 20
+ 2
40
16
609
+ 49
+ 3
147
17
639
+ 79
+ 4
316
18
619
+ 59
+ 5
295
19
658
+ 98
+ 6
588
20
631
+ 71
+ 7
497
21
683
+ 123
+ 8
984
22
710
+ 150
+ 9
1,350
23
710
+ 1.50
+ 10
1,500
24
721
+ 161
+ 11
1,771
25
718
+ 158
+ 12
Sum: 2
1,896
= 18,276
166 HOMOGRADE STATISTICAL SERIES. [93
nary manner we find:
indicating an increase of death from cancer of about 14 persons
pr. annum for a population of 1,()0(),0()(). Eliminating the secular
disturbances in the same manner as above, we now get a coefficient
of disturbancy equal to ().9S3/ (/ = V — 1), practically a normal
dispersion when taking into account the mean error due to
sampling.
93. Application of the Lexian Dispersion Theory in Actuarial
Theory. Conclusion. — The Russian actuary, Jastremsky, has
applied the Lexian Dispersion Theory in testing the influence of
medical selection in life assurance.^ The research by Jastremsky
evolves about the following question. Is medical selection a
phenomena independent of the age of the assured? Let ^'^(/^
denote the observed rate of mortality after t years' duration of
assurance. In the same manner q/'^ denotes the rate of mor-
tality of a life aged x after 7 or more years of duration (i ^ 5).
Forming the ratio ^'V/^ : f/^^^^ for all ages of x we obtain a certain
homograde series for which we may compute the Lexian Ratio
and the Charlier Coefficient and thus determine if the fluctuations
are due to samphng onlv or dependent on the age of the assured.
Space does not allow us to give a detailed account of the very
interesting research by Jastremsky as applied to the Austro-
Ilungarian Mortality Table (Vienna, 1909), and we shall limit
ourselves to quote his final results as to the Lexian Ratio, L,
for Whole Life Assurances and Endowment Assurances:
Whole Life Assurances.
Endowment Assurances
t
L
L
1
0.88
1.01
2
0.89
0.96
3
1.12
1.05
4
1.05
0.98
5
1.07
0.91
The above values of L all lie close to unity and the series may
therefore be considered as a BernouUian Series where the fluctu-
' Ja.strcmsky. "Dcr Aualesc-Koeffizient," Zcilschr. f. d. yes. Vers.-Wiss.,
Band XII, 1912.
93 ] APPLICATION OF THE LEXIAN DISPERSION THEORY. 167
ations are due to sampling entirely. Or in other words, the ratio
iPt = ^'^7^ : qJ'^^ is a quantity independent of the age of the
assured.
The great majority of statistical series may be subjected to a
similar analysis as given in the preceding chapters. The char-
acteristics as described previously, the Lexian Ratio and the
coefficient of disturbancy, tell us the magnitude of possible fluc-
tuations from sam])le to sample. In many cases we may by means
of the secular coefficient of disturbancy, ^, partly or wholly
eliminate such fluctuations, due to secular causes, and thus be in
a better position to study the periodical fluctuations.
A statistical research may be likened to the navigation of a
difficult waterway, full of hidden rocks and skerries out of sight
to the navigator. The amateur statistician, sailing the ocean
in a blind and happy-go-lucky manner, often comes to grief on
those rocks and suffers a total shipwreck. The skillful navigator,
the mathematically trained statistician, is always on the lookout
for the sea marks. In the Lexian Ratio and the Charlier Coef-
ficient of Disturbancy he recognizes a beacon light, often signal-
ling "Danger ahead." He stops his engines. In case he does
not possess the particular charts giving the exact location of the
hidden reefs his prudence advises him to call a i)ilot to bring his
ship safely in harbor. On the other hand, if he has reliable
charts and knows his profession thoroughly he may venture
forth and do his own piloting, by a study of the charts. It is
to the study of such charts — i. e., a special study of the higher
statistical characteristics — that we shall turn our attention in
the second i)art of this treatise. The reader who has followed us
up to this point may perhaps feel discouraged by realizing how
little he has gained in knowledge after having learned a mass of
technical detail and formulas. We can quite appreciate and
understand this feeling. So far, he has perhaps chiefly been
impressed by the treacherous and misleading character of sta-
tistical mass phenomena, but to recognize a danger signal and
thus avoid the pitfalls is one of the fundamental essentials in
safe navigation in statistical research.
PART II
FREQUENCY CURVES AND
HETEROGRADE STATISTICS
CHAPTER XIII.
THE THEORY OF ERRORS AND FREQUENCY CURVES AND ITS
APPLICATION TO STATISTICAL SERIES. HISTORICAL NOTES.
94. General Remarks. The Hypothesis of Elementary Er-
rors. — In the previous chapters we have discussed the elementary
statistical parameters, the mean and the dispersion, together with
the Lexian ratio and the Charlier coefficient of disturbancy and
their application to the mathematical analysis of the homograde
series. We shall now extend this discussion to the parameters of
higher orders, such as the skewness and the excess, and also give
the theory for a mathematical analysis of the other great domain
of statistical series, the heterograde series.
The main reason for the separate treatment of the homograde
statistical series is on account of their close analogy to ordinary
mathematical probabilities. Whenever the number of comparison,
s, may be regarded as equivalent to the total number of equally
possible cases in ordinary a priori probabilities and the observed
occurrences of the attribute (event) as the favorable number of
cases, m, among the total number of possible events, s, we are
justified in regarding the ratio ni : s in the light of a mathematical
probability. For this reason all homograde series may be ex-
plained as being sul)ject to the same mathematical laws as those
governing ordinary a priori probabilities, which are fully explained
by the series of Bernoulli, Poisson and Lexis and the various
combinations of such series. IMoreover, in all such series it is
possible to compute both the mean and the dispersion by the
indirect or combinatorial process instead of the direct or physical
process.
The nucleus of the three fundamental series, the Bernoullian,
the Poisson and the Lexian, as well as their various combinations
is found in the development of the point binomial (p + qY of
the Bernoullian Theorem as described in Chapter IX where the
general term expressing the probability of the occurrence of an
event E a times and of the complementary event E, ^ = s — a
times is given by the formula
169
170 THEORY OF ERRORS AND FREQUENCY CURVES. [ 94
The numerical computation of this exact expression becomes,
however, too unwieldy for large values of s and we shall therefore
try to replace it with a more flexible approximation, preferably
by a continuous function or by a rapidly convergent infinite series.
On page 101 we gave such an approximation formula for the
maximum value of <p (a), denoted by the symbol Tm- We wish,
however, to find a simpler expression for the more general term as
well. This further development necessitates the determination of
several higher characteristics or parameters than those expressed
by the mean and the dispersion. If we should succeed in this task
the homograde series can be fully explained by the theory of
mathematical probabilities and placed upon a sound a priori
basis.
The question now arises whether the a priori theory of mathe-
matical probabilities will furnish a similar basis for the second
domain of statistics, the heterograde series. We are of course able
to compute by means of the direct or physical process both the
mean and the dispersion in various heterograde series, such as
measurements on heights of adult males, number of fin rays in
fishes or number of telephone calls over a trunk line in a given
interval of time. But are we also able, like in the case of the
homograde series, to forecast those parameters by means of the
criterions of the Bernoullian, Poisson and Lexian series, i. e. by
the indirect or combinatorial process?
A simple consideration will soon lead us to the admission that
no a priori reasoning or a simple theorem like that of the Ber-
noullian will enable us to forecast the mean stature of Danish,
Norwegian, Swedish or English adult males or the mean number of
telephone calls over a trunk wire in a given interval of time. And
while we by the physical or direct process can compute both the
mean and the dispersion from previously collected statistical data,
we have no way of knowing whether such parameters, purely
empirical in form and nature, have any real significance beyond
that of abstract mathematical calculations. Nor do such empirical
parameters offer similar explanations as those of the homograde
series. We arc for instance able to predict the probability that in a
series of 1000 successive drawings (with replacements) from a
deck of whist the number of drawn aces will fall between 100 and
120. But we are not able by an a priori reasoning or by mathe-
matical deduction to forecast the probability that among 1000
Scandinavian adult males — all chosen at random — the height
94 ] GENERAL REMARKS. 171
of an arbitrarily selected individual will fall between 170-175
centimeters.
Experience has, however, shown that the heterograde series
show similar grouping tendencies around the mean value as those
encountered in the homograde series. As an example we may-
compare the BernouUian series of black cards in sample sets of 10
as collected by M. Charlier and shown on page 138 and the Poisson
series of black l)alls in sample sets of 20 collected by Mr. Bonynge
(shown on page 143) with a series of measurements, relating to the
heights of Danish conscripts for the year 1916. Such a comparison
is given below.
Charlier's Bonynge's Danish Anthropometric
Data Data Data
m X F{x) m x F{x) w x F{x)
0— .5 3 0—5 2
1
— 4
10
2
— 3
43
3
— 2
116
4
— 1
221
5
247
6
+ 1
202
7
+ 2
115
8
+ 3
34
9
+ 4
9
10
+ 5
1
— 4
9
2
— 3
35
3
— 2
52
4
— 1
86
5
109
6
+ 1
85
7
+ 2
69
8
+ 3
30
9
+ 4
16
10
+ 5
6
11
+ 6
1
140-145 cm.
— 5
32
146-150 "
— 4
44
151-1.55 "
— 3
243
156-160 "
— 2
1284
161-165 "
— 1
3777
166-170 "
5742
171-175 "
+ 1
4796
176-180 "
+ 2
2129
181-185 "
+ 3
588
186-190 "
+ 4
81
over 191 "
+ 5
11
1000 18727
500
The grouping tendency or the clustering around the mean value
is manifest in all three series; but while this tendency in the case
of the two homograde series as offered in the experiments by
Charlier and Bonynge may be fully explained by means of the
theorems of mathematical probabilities no such reasoning is
sufficient to explain the clusteruig tendency of the heterograde
series relating to Danish conscripts. The calculus of probabilities
in itself would not be sufficient to explain the grouping tendency of
the variates in a heterograde series unless a general hypothesis
will aid us in explaining the variation among several heterograde
objects in respect to a specific attribute.
Thus the question which now confronts us is whether it is
possible to establish a simple hypothesis which will enable us to
extend the principles of the mathematical theory of probal)ilities
172 THEORY OF ERRORS AND FREQUENCY CURVES. [ 94
to the domain of the heterograde series and to build up a theory
similar to that of the honiograde series. The great Laplace was
the first to solve this problem, and his investigations and analysis
on this important subject are indcHnl some of the most important,
but also some of the most difficult to follow in his Theorie des
Probahilites. The hypothesis employed by Laplace in explaining
the phenomena of variation in a heterograde series is the hypothesis
of elementary errors. The hypothesis was later on somewhat
simplified by the German astronomer and engincM^-, Hagen, and
it has of late years been further developed through the elegant
researches of the Scandinavian astronomer and statistician,
M. Charlier. According to the Laplacean— Hagen — Charlier
theory every variate or individual deviation from a certain norm is
generated as the sum of a mass of small and unknown quantities —
generally infinite in number — which are known as elementary errors
(deviations) . The word error nuist of course be tak(Mi in a different
sense than that we usually associate with the word. In precision
measurements we are actually dealing with true or natural errors
arising from impcu-fections of iho instruments and the observer,
but it would of course not be right to regard a deviation of say
5 centimeters from the mean stature of a population group as an
error in the usual sense of the word. Used in its wider sense as an
expression for deviations the term will, however, be readih' under-
stood, and it is in this sense we shall use it in the following jiages.
Expressed in matheniatical symbols the hypothesis of elemen-
tary errors may be presented as follows. Let .r^. (where k =
1, 2, 3, . . . s denotes the kth error source among a total of s
sources) represent the magnitude of a statistical variate expressed
as a deviation (error) from a certain norm, then
/, (r) (A; = 1, 2, 3, . . . s)
may be regarded as the probability that x^ assumes the value r.
As to this particular elementary error probability function Laplace
makes no other assumptions than those which follow directly from
the definition of a mathematical probability. That is to say
0</. (r)<l,
where A; = 1, 2, 3, . . . s and r = ± 1, ± 2, ± 3, . . . d= co
Since it is certain that one or more of the aljove values of r,
95 ] APPLICATION TO STATISTICAL SERIES. 173
whether positive or negative, are bound to occur, we have evidently
the relation
r = +00
The epoch-making analysis of Laplace lies in the determination
of the unknown function /^(r) from such simple and general
assumptions.
95. Application to Statistical Series. Definitions. — The La-
placean-Charlier hypothesis of elementary errors opens the way
for a mathematical analysis of a vast number of statistical data
and series, which we shall briefly discuss in the following para-
graph. First of all we submit therefore the following definition of a
statistical object.
A number of similar objects (a species) which can be arranged in
numerical order according to the measurable variation of a certain
observed attribute (character), also called a variate, is known as a
statistical object, eventually as a statistical series.
It is readily seen that this definition covers a wide range of sub-
jects and that statistical methods instead of being applicable to
social and economic problems only are equally useful in botany,
zoology, biolog>' and even in astronomy, physics or chemistry.
Moreover, since the deviation of an individual variate of the
statistical series as measured from a certain arbitrarily choosen
norm evidently may be regarded as the sum of several elementary
errors (the word error to be taken in its wider sense), it is evident
that the statistical object can be subjected to a mathematical
analysis on the basis of the theory of errors.
A simple consideration will also convince the reader that the
above definition covers not alone the heterograde series but also
the homograde series. For instance in the Bernoullian and Poisson
series as presented in the experiments by Charlier and Bonynge
on page 170, the number tn, which gives the number of favorable
events in each sample set, may be considered as a statistical
variate and F{x) as a statistical series.
This simple fact is of the utmost importance since it makes
it possible to treat both the homograde and heterograde
series on the common ])asis of elementaiy errors and links
in the case of the homograde series the a priori mathematical
probabilities with the a posteriori probabilities. Such connection
174 THEORY OF ERRORS AND FREQUENCY CURVES. [ 95
is of special interest in the further treatment of the celebrated
Rule of Bayes.
While thus the honiograde and heterograde series may be
viewed from a common viewpoint it is, however, necessary to
point out a distinct difference in the nature of the statistical
variates themselves. In one case we find the variate (the measura-
ble attribute) expressed in whole numbers onlj', such as the number
of fin rays in fishes, petals in flowers or the occurrence of a specified
color in card drawings. The variates are in such cases known as
integral variates. The observations on tail fin rays of flounders by
the Danish biological station on page 131 offer such an example.
As a further illustration of integral vjjriates we choose the follow-
ing statistical series from the observations of the English phys-
icists, Rutherford and Geiger. Messrs. Rutherford and Geiger
counted the numbers of alpha particles radiated from a bar of
polonium during a long series of intervals, each lasting one-eighth
of a minute. The table states the number of times, F(x), the
number of particles omitted in this interval had a given' value, x.
X
F{x)
57
1
203
2
383
3
525
4
532
5
408
6
273
7
139
8
45
9
27
10
10
11
4
12
13
1
14
1
(
As an example of a very slight variation the Danish biologist and
botanist, W. Johannsen, quotes the following observations by
his colleague Professor Raunkjcer of Copenhagen on the number of
involucral leaves of 100 samples chosen at random of taraxacum
erythrospernum:
No. of Leaves Frequency
X F{x)
13 99
14 1
95] APPLICATION TO STATISTICAL SERIES. 175
In other cases it is not possible to express the measure of the
attribute in whole numbers. Thus measurements of stature,
chest circumference and weight of recruits, or measurements of the
percentages of sterility in wheat, barley, r3'c and oats will in
general possess all possible fractional values between two integral
numbers. Hence we must group the; observations in classes, and
such classified variates are known as graduated variate.s.
The measurements of heights of Danish conscripts for the
year 1910 and shown on page 170 offer an illustration of grad-
uated variates. Another case is furnished in the number of deaths
by attained ages in a mortality table. In most mortality tables
the deaths are given by integral ages only and represent then^fore
strictly sp(>aking integral variates.
In biology we encounter numerous homograde series especially
in investigations on dimorphism or polymorphism. Johannsen of
Copenhagen produced from crossbr(>eding between a species of
beans with white blossoms and yellow seeds and a species with
violet lilossoms and black seeds a bastard species with violet
blossoms and muddy colored seeds. The offsprings — 558 individ-
ual plants — of this Ijastard showed following variations :
White Blossoms Violet Blossoms
IGO 398
Color of Seeds Color of S<'eds
I I
II II
yellow bronze violet black
39 121 105 293
In respect to blossoms we have two alternatives, in respect to
seeds 4 alternatives.
As a few illustrations of the wid(^ range of vai'iable phenomena
which allow to be classified as statistical objects, we present the
following table:
176
THEORY OF ERRORS AND FREQUENCY CURVES.
96
Statistical Object.
Sample.
Attribute.
Variate.
Integral.
Seriefi.
Drawings of diamonds
from deck of card.
Series of s
drawings.
Frequency of
Diamonds.
Homograde.
Petals of Flowers.
Individual
flowers.
No. of Petals.
u
Heterograde.
Fin Rays of Fish.
Individual
fishes.
Xo. of Rays.
"
II
Sex at Birth.
Series of s
births.
Frequency of
Male Births.
u
Homograde
Age Distribution of
Assured Persons.
The Policy-
holder.
Age.
Graduated.
Heterograde.
Distribution of
Amounts Assured.
The Policy.
.\mount in
Dollars.
"
11
Antropometric Meas-
ures of Recruits.
The Recruit.
Chest Measure.
II
Precision Measures.
Individual
Measure.
Error.
"
"
Invalidity.
The Individ-
ual Pa-
tient.
Duration in days
of Invalidity.
-
"
Income and Wages.
The Individ-
ual
Worker.
Yearly Income
in Dollars.
Cross of Flowers.
Series of s
Hastard
Flowers.
Color of Blos-
soms, Seeds,
etc.
Integral.
Homograde.
It is to be noted that in the primary observations on homograde
series the num])ers are all abstract, whereas the heterograde series
consist of concrete numbers. Another peculiarity of the homo-
grade series Is that they are always connected witli the number of
comparison, .s, which is absent in the heterograde series.
96. Compound Frequency Curves. — According to the Lai)laccan-Charlicr
hyT)othcsis any fncjucncy curve may be considered as being generated as a
sum of independent frequency curves and represents therefore in the final
96] COMPOUND FREQUENCY CURVES. 177
instance really a compound frequency curve. Mathematically speaking wo
may therefore consider any frequency curve, no matter of what form, as
being represented by the symbolic relation.
Fix) = 2 Niifiix) for i = 1, 2, 3, . . .
The functions (fiix) may sometimes all be normal or Laplacean probability
curves. On the other hand, by assigning different values to Ni or the areas
of the separate curve we may ol)tain a (!omi)ound curve of wavelike form.
Suppose for instance we had two samples of <)l)servations on the heights of
say 100,000 Japanese recruits and 10,000 Danish recruits, eacih individual's
measure written on separate cards. Suppose furthermore that we mixed those
110,000 cards in an urn, and then formed a new frequency distribution of this
mixture This new frequency distribution would be a compound curve with
two strongly }m)nounced crests or maximums. One (the Japanese) clustering
around the value of 160 centimeters, while a smaller crest (the Danish) would
tend to cluster around the value of 170 centimeters.
Another instance is offered in the distribution of the frontal breadths of a
Naples specimen of the crab, Carcinus mcvnas, as measured by Weldon. Wel-
don thought it very probable that this rather skew frequency distribution
was produced by a fusion of two distinct races or species of individuals, which
were clustered symmetrically around separate means. The distinguished
English biometrician, Karl Pearson, tested this hypothesis for him and
analysed the compound curve as two component curves representing respec-
tively 58.55 and 41.45 per cent of the total area of the comj)omid frequency
curve. Thus Weldon's hypothesis was verified by a mathematical analysis.
A quite different type of example is offered in the frequency distribution of
deaths by attained age as represented by the dj. (!olumn in any ordinary mor-
tality table. The fact that the deaths in th(> dj^ column showed a marked
clustering tendenc-y, strongly suggestive of the normal Laplacean curve around
the age group 70-75, was already noted by Lexis, who in this way made a
very interesting attempt to determine what he called a Normalalter for
the age of man. Later on Italian statisticians took up the problem and ana-
lysed the dx curve of Italian life tables as a sum of several normal frequency
curves. Karl Pearson was the third investigator to take up the problem in a
very fascinating essay in his Chances of Deaih. Pearson pictures Death as 5
marksmen shooting at a human target passing over the Bridge of Life. Each
marksman aims with different precision and skewness. The result is 5 com-
ponent skew curves.
Although the brilliant and perfect literary style of the eminent English
biometrician rouses the admiration and brings forth the reader's unstinted
praise, I can, however, not help being in accord with the distinguished Amer-
ican biologist and statistician, Raymond Pearl, who in his 1920 Lowell Lec-
tures, although speaking in tlu; highest terms of praise of Pearson's work,
characterized a mathematical analysis of this kind as being nothing more than
a highly inter(>sting and neat graduation foriiuila Init wholh' void of any
biological significance.
It is as a mere matter of fact a comparatively easy matter to break np any
death curve of a mortality table into separate mathematical components. As
an example of such a process I offer the illustration in Chapter XVI, where
178 THEORY OF ERRORS AND FREQUENCY CURVES. (97
I havo broken up the recently published American AM^^^"* mortality table
into tAvo curves of the Gram-Charlier type, obtaining as g;ood results as the
Italian investigators and Pearson, who use 5 component curves.
But as already pointed out by Lexis "a mere mathematical analysis in
component groups does not enlarge our knowledge of the causal relationships.
It would be a quite different matter, however, if it were possible to establish
clustering tendencies around definite ages for each of the more important
causes of death."
An attempt to do this has been made by the present \\Titer in his forth-
coming book An Elementary Treatise on Frequency Curves and their Applica-
tion to the Human Death Curve. I start with the hypothesis "that the frequency
distribution of deaths at attained ages classified according to certain groups
of causes of death among the survivors in a mortality table tend to cluster
around specific ages in such a mamier that their frequency distribution can
be represented by a Gram-Charlier frequency cui-ve." If this hypothesis can
be accepted as having a sound biological basis I have shown that it is possible
by a mathematical analysis resting on such hypothesis to construct mortality
tables from mortuary records by sex, attained age and cause of death, and
vrithout any information about the number of lives exposed to risk at various ages.
This j)roi)osal has been met by a storm of j^rotest from many American ac-
tuari(\s, who claim that 1 have attempted the impossible. Final judgment
should be suspended, however, until the actual appearance of the work, which
I think must be judged from a biological rather than from a mathematical
point of view. The fact that the method has given good results in the con-
struction of many mortality tables among highly different races and occupa-
tions must, I think, be attributed to purely biological causes and not to ac-
tuarial or mathematical methods, which in the process have been employed
as a mere tool, as a means rather than as an end.
97. Early Writers. — -The idea of frequency curves or frequency
distribution is probably very old. It very likely aros(^ in the mind
of man when he began to make quantitative observations. Un-
doubtedly the surveyors and engineers of the pcoi)le of ancient
civilization had noticed that successive and independent measure-
ments of the same object often showinl variations. On the other
hand we have no means of knowing if the ancient geometers and
mathematicians knew how to estimate and value such variations
from the true value of the object. It is probable that the great
Greek astronomers, such as Hipparch antl Aristarch in their
astronomical observations have employed some rational method
of allowing for errors due to th(^ instruments and the individual
observer, but no records are availal)le so as to settle this question.
The great Danish astronomer, Tycho Brahe, the fatlu^r of mod-
ern astronotny, on the other hand made careful adjustments for
errors of obs(;rvations and has left us records on the systematic
method of such adjustments.
98 ] LAPLACE AND GAUSS. 179
However, it was not before the close of the eighteenth century
that the errors of observations were subjected to a mathematical
trcat'ucnt. The first known writer on the mathematics of the
subject was the English actuary, Thomas Simpson, a most remark-
able self-taught mathematician who in 1757 issued his "Miscel-
laneous Tracts on some curious and very interesting subjects in
Mechanics, Physical Astronomy and Speculative^ Mathematics."
In this interesting and instructive little book is found a chapter
entitled "An Attempt to shew the Advantage arising by Tak-
ing the Means of a Number of Observations in Practical As-
tronomy."
About 15 years later the French mathematician, Lagrange, took
up the ideas of Simpson in a memoir, which at the time caused
considerable notice in mathematical circles. Lagrange in his
treatment followed a course very much similar to that employed
by de Moivre in the discussion of the problem which bears his own
name.
In 1778, Daniel Bernoulli in the scientific publications of the
Russian acadeny of Petrograd subjected the memoirs of Lagrange
to a searching criticism and proposed the first mathematical
formula for a frequency curve or curve of errors around the mean.
Bernoulli suggested as a law of error or frequency function, (p (a:),
the following expression:
^(.r) = + \/ r — x\ where /• is a constant
This (Hiuation rei)resents a symmetrical semi-circle and gives as
we shall hav(> occasion to show at a later stage a rough approxitna-
lion to tlie presumptive law of erroi'.
A very important c()ntri])ution to the theor}^ was also made
by the American, Adrain, in his journal "The Analyst."
98. Laplace and Gauss. — Laplace was the next mathematician
to take up the subject of fretiuencv curves in his monumental
work "Theoi-ie analytique d(\s probabilites." The great French-
man dealt with the subject in a manner which leaves little to
be desired. ]\I. Charlie^-, the eminent Swedish astronomer and
statistician, has justly remarked that among the various deductions
of the law of errors, the exhaustive researches of Laplace occupy
beyond doubt a leading position because of their generality and
far reaching applications. On the other hand, the analysis of
Laplace is by no means easy to follow in all its details and the
4th chapter of the "Theorie Analytique des Probabilites" accord-
180 THEORY OF ERRORS AND FREQUENCY CURVES. [ 98
ing to a remark by Todhunter in his "History of Probabilities"
forms one of the most important but at the same time also one of
the most cliffieult parts of the great work.
No (ioiil)t the extreme difficulty of fully mastering the far
reaching l)ut intricate analysis of Laplace was realized by the
mathematicians. Already his friend and disciple, Poisson, realized
this and issued in 1832 a note entitled "Sur la probabilite des
resultats niovens dvs observations." But the wealth of ideas in
Lajilace's treatise and their wide range of application were really
never fully recognized for almost a full century when they were
taken up by Charli(>r, who more than any one else has proven
their gr(^at worth as the most general and direct basis for a com-
plete theory of frequency functions and the associated problem of
correlation.
In the meantime Laplace's method had l)een supplanted by the
independent and contemporary researches of the great German
mathematician, Gauss. The method employed by Gauss in
deriving his law of error or freqiu^ncy curve and the therewith
associated crit(M-ions of the method of least squares is undoubtedly
very simple and (>legant and nmch easier to follow for the beginner
than the analysis of Laplace. Gauss in his stutlies confided himself
to the so-called precision (>rrors or errors arriving from repeated
measurements by means of physical instruments, such as astronom-
ical or geodetic observations or measurements in experimental
physics or chemistry.
The ideas put forth by Gauss were followed up by a number of
astronomers and [)hysicists, such as Bcssel, Encke, Hansen, and
Hagen of Germany, Andrae, D'Arrest, and Gylden of Scandi-
navia, Airy, Herschell, and Tait in England, Laurent in France,
and Newcomb and Chauvenet in America. And the Gaussian
UK^hods ai-e still used exclusiv(4y in preference to those introduced
by Laplace in most of our text-books on theoiy of errors and the
related sul)ject of l(>ast squares.
One reason for this preference for the theory of Gauss apart
from its simpUcily of representation is to be look{Hl for in the fact
that until a compaiatively recent date the majority of applications
of the theory of fi(>(iuency curves or error curves had reference to
precision measurements. As pointed out by N. 11. Jorgensen,^ in
his excellent Danish tn^atise on "Frequency Surfaces and Coi-rela-
tion" it will, as a I'uie, be found that the Gaussian (>rror law may be
1 Undersiigclscr over IVoquonsfladcr og Korrelation (Copenhagen, 1910).
99 ] quetelet's studies. 181
regarded as an excellent method of approximation, which becomes
well-nigh perfect in the case of errors of precision measurements
with dc^licate instruments in the hands of carefully trained ob-
servers. The Gaussian frequency curve may therefore be said to
fulfill all the requirements in pli-axi s of a law of eri-or, where we are
concerned with errors in the true sense of the word.
99. Quetelet's Studies. — Matters became, however, quite differ-
ent when the biologists and economists began to employ math-
ematical analysis in their research work. It was the great Belgian
astronomer and statistician, Quetelet, who first introduced exact
measurements in the study of biological and anthropological
phenomena and showed that a number of collected statistical data
on heights, weights and chest measurements of recruits exhibited a
close conformity to the Gaussian law of error, although the varia-
tion among the individual objects as measured could not be con-
sidered solely as erroi's in the original sense of the word.
Investigations along this line were greatly accelerated by the
discoveries of Quetelet. All sorts of measurements were taken and
the rapidly growing collections of statistical data relating to
economic and social conditions as recorded by various govern-
mental statistical bureaus furnished material for further investiga-
tions. But unfortunately in all these investigations the Gaussian
error law came to act as a veritable Procrustean bed to which all
possible measurements should be made to fit. The belief in
authority so typical of modern German learning and which had
also spread to America was too great to question the supposed
generalit}' of the law discovered by the great Gauss. Statisticians
could not conciliate themselves with the thought of the possible
presence of "skew" frequency curves, although numerous data
offered complete defiance to the Gaussian dogma and exhibited a
markedly skew frequency distribution. Supposedly great author-
ities argued naively that the reason the data did not fit the curve
of Gauss was that the observations were not numerous enough to
eliminate the presence of skewness. In other words, skewness was
regarded as a by-product of sampling and was believed could be
made to disappear completely if we could take an infinite number
of observations.
Voices had, however, been raised against these energetic but
futile Procrustean efforts. Already Quetelet realized the existence
of skew frequency curves. This is clearly brought out in his
correspondence on this subject with Mr. Bravais of the Ecole
182 THEORY OF ERRORS AND FREQUENCY CURVES. [ 100
Polytechnique of Paris as published in an appendix to his "Lettres
siir la Th('oi-i(> (l(^s ))i'ol)abilites."
100. Opperman, Gram, and Thiele. — Neither Quetelet nor
Bravais succeeded, however, in giving a complete mathematical
treatment of the theory of skew frequency curves. The first com-
plete mathematical demonstration of this aspect of the matter was
given by various Scandinavian investigators. A Danish actuary,
Opperman, was pi'()l)ably the leading spirit in organizing the
r(>volt against the i)elief in authority as preached by the adherents
of the doctrine of Gauss. Opperman, who was a self-taught
mathematician, seems to have looked with suspicion on many of
the researches b}^ German mathematicians of the latter half part
of the nineteenth century. He was a great admirer of the early
Scotch and English mathematicians with whose works he was
thoroughly familiar, and it is said he took great delight in pointing
out how many of the lengthy and formidable German demonstra-
tions in the realm of the theory of functions had been demon-
strated in a more elementarj' and clearer manner b}' such men as
Wallis, Stirling, MacLaurin, Gregory, Briggs and Napier. As a
practical actuary and managing din^ctor of the Danish Govern-
ment Life Assurance Fund he had ample opportunity to notice
that many frequency distributions occurring in actuarial work
offered a notorious defiance to the frc^quency curve of Gauss.
Around Opperman there gathei-ed a small gi'oup of young en-
thusiastic students of mathematics among whom we may espe-
cially mention Gram and Thi(>le and to whom he expounded his
ideas. Opperman himself wrote very little and alwa\'s in a con-
densed form. A reviewer of his work remarks that he rewrote his
essays several times so as to be able to represent on a single page
what other mathematicians usually required a dozen of pages to
express. He has left very little material bearing on the theory of
frefiuency curves, but his discussions on this subj(>ct with his
younger disciples evidently bore fruit.
J. P. Gram was the first mathematician to show that the normal
symmetrical Gaussian error curve was but a special cas(> of a more
general system of skew frequency curves which could be repre-
sented by a series. In his very original doctor's thesis in Danish
on "The development of series by means of the method of least
squares" (Copcnhagan, 1879) ^ he extended some theories orig-
inally expounded by the Russian mathematician, Tchebycheff, to
' Om Raekkeudviklinger.
100] OPPERMAN, GRAM, AND THIELE. 183
the representation of frequency functions by means of a series.
By using the Gaussian curve
- (X- My
20-2
as a seneratinp; function Gram showed that an arbitrary frequency
function could l^e represented approximately by a series of the form
F(X) = C,<p (X) + Ci (p' (X) + Co (p" (X) + C3 (p'" (X) + C4 <p"" (x)
+ . . .
In this development Gram established some far reaching prop-
erties of infinite determinants and their relations to orthogonal
functions which later have become of much use in the recent epoch-
making researches on integral equations b3^ the Swedish math-
ematician and actuary, Fredholm.
To Gram, therefore, belongs the honor of having been the first
mathematician to give a systematic theory for the development of
skew frequency curves.
While Gram's later work as a managing tlirector of a life insur-
ance company occupied most of his time and left but little o[)por-
tunity for purely mathematical work his friend, T. N. Thiele, began
to lecture at the University of Copenhagen on the general theor}' of
observations. The substance of these decidedly original lectures
was in 1889 published in book form under the title "A General
Theory of Observations." -
In several respects this woi-k occupies a dual position to the
work of the great Laplace, although Thiele is set like flint against
the idea of basing the theory of probabilities on the conception of
an a priori probability. In his lectures he always maintained that
the greatest benefit derived from the study of the method of least
■ square was that the student learned where not to use it. Among
one of the great achievments of Thiele is the introduction in the
theory of frequency curves of a certain system of statistical char-
acteristics to which he gave the name of semi invariants and which
are practically identical to the system of moments later on intro-
duced by Pearson. By means of these semi invariants Thiele
arrived at the same series as deduced by Gram. In Thiele's work
we also find a very original treatment of the theory of correlation
- Almindelig lagttagelseslsere.
184 THEORY OF ERRORS AND FREQUENCY CURVES. [ 101
originally introduced by Bravais. But instead of the term correla-
tion he uses the words "bonded observations."
Like Laplace's "Theorie des Analytiques des Probabilites "
Thiele's original work and its subsequent abridged translation in
English offers by no means an easy reading, especially to the
beginner. It contains, however, like the work of Laplace a verita-
ble wealth of ideas and methods which remain unsurpassed in the
realm of mathematical statistics and no serious worker on the
general theory of observations can afford to neglect to study the
original works of Laplace and Thiele.
101. Modern Investigations. — The investigations of Gram and
Thiele bring us up to the close of the nineteenth century. Their
ideas reached but a small number of students of mathematical
statistics because of the very limited knowledge of Scandinavian
languages among mathematical readers in general. But from the
beginning of the nineties other voices began to be heard against
the Gaussian dogma. In Germany it was Fechner who first
entered the ranks of the opposition with his so-called "zweispa-
Itiges Gesetz." His work was continued by Lipps and the Leipzig
astronomer, Bruhns, who by the publication in 1906 of his " Kollek-
tivmasslehre " gave an almost complete theory of frequency curves
where we again find the series originally developed by Gram and
Thiele.
Although quite considerable valuable work has thus been done
in Germany along the lines of frequency curves it was, however,
in England that the renewal of the classical probability theory
took place with the renowned memoirs by the English math-
ematician, Karl Pearson, entitled "Contributions to the Math-
ematical Theory of Evolution" in Philosophical Transactions for
1895.
Since that year Pearson has produced a certain type of statistical
literature, of almost Gargantuan proportions. The quarterly
journal "Biometrika" of which he is the editor is devoted to the
mathematical study of biological problems. When Pearson first
introduced his famous types of curves (now more than 12 in num-
ber) the study of frequency distributions was greatly accelerated.
The application of these curves to biological i)roblems was ap-
parently so simple that they were used in a rather loose manner
by many biologists and anthropologists who had but little training
in mathematical analysis. Examples of such pseudo mathematical
analysis are especially found in the writings of the American
101 ] MODERN INVESTIGATIONS. 185
anthropologist, Franz Boas, which may be held up as a warning
to all statisticians to keep away from the higher mathematical
analysis of collected statistical data unless they are familiar with
the tools of the probability calculus.
Such misuse can of course not be laid at the door of Mr. Pearson
who indeed has protested vigorously against the erroneous appli-
cation of his methods by investigators of the Boas type. On the
other hand, it is equally true that Mr. Pearson at times has relied
too nmch on his mathematical formulas and violated the maxim
of the Danish biologist, Johannsen, that "we must practice
biology with mathematics and not as mathematics."
The immense production of Pearson coupled with his well-nigh
perfect and forceful style of writing has to a certain extent over-
shadowed the researches of his compatriot, Edgeworth, whose
works according to the Danish actuary and mathematician,
Jorgensen, are greatly superior to those of Pearson both in scien-
tific rigor and in practical applications. Edgeworth has deduced
the previously mentioned series by Gram and Thiele in a very
elegant form in the Cambridge Philosophical Transactions (1904)
and he has in a series of articles in the Journal of the Royal Statis-
tical Society outdistanced many of his contemporaries among
the mathematical statisticians. Unfortunately Edgeworth's con-
tributions have not gotten the attention they deserve, probably
because of the rather fragmentary and unsystematic maimer in
which they have appeared. Among the new methods introduced
by Edgeworth we may particularly mention the so-called method
of translation.
The Pearsonian types of frequency curves are represented by
formulas which in mathematical language are termed closed
expressions in contradistinction to the development in series. This
latter method is still being preferred by the Scandinavian math-
ematicians under the leadership of the Swedish astronomer Charlier.
Charlier started his first investigation with a small brochure en-
titled " Uberdas Fehlergesetz " in the Meddelendan for 1905, wherein
he followed the method originally introduced by Laplace and in a
most elegant way of. deduction reached the series of Gram and
Thiele. He has since that year published a series of small mono-
graphs on various aspects of mathematical statistics and their
.application to stellar statistics which beyond doubt are destined to
become classics in the history of probabilities.
Charlier has shown that all frequency curves fall into two types
186 THEORY OF ERRORS AND FREQUENCY CURVES. [ 101
which he has designated as type A and type B. The A type is the
usual expansion of Ciiani and Thiele with the normal frequency
curve as the seneratin<i!; function. Type B which covers decidedly
skew frequenc}' curves is given by the series
Fix) =Coi^ (.r) + 1^' A (^ (.t) + I A^ <p {x) -{- . . .
( - 1)"
where
X
- A /» XcosOO
<fj (x) = — j c COS [Xsinco — xo}]dcji)
TT J
is the generating function.
The decidedly constructive work begun by Charlier has been
ably supplemented by his talented disciple Wicksell and the
Danish actuary, Jorgensen. Wicksell in 1C20 issued in Swedish a
series of lectures on mathematical statistics delivered during the
autumn of 1919 before the Swedish Assurance Society. He has
also written numerous excellent monographs on mathematical
statistics and their application to vital statistics.
N. R. Jorgensen issued in 191G his large octavo volume on
"Researches on Frequency Surfaces and Correlation" ' which
beyond doubt is the most important work among the contributions
of Danish actuaries since the appearance of the memoirs of Gram
and Thiele. Jorgensen's systematic treatise has greatl}' furthered
the studies of the Scandinavian school both in tlu^oretical and
practical aspects. A very important feature of his book is the
insertion of an (extensive collection of numerical tables of various
functions which greatly facilitates the jiractical api)lications of the
theory. These tables, many of which are the results of his individ-
ual efforts, hold equal rank with the well-know^i ''Tables for j
Biometricians and Statisticians" edited by Karl Pearson in 1914. ^
Besides the writings of Charlier, Wicksell and Jorgensen a
number of Scandinavian mathematicians, actuaries and statis-
ticians have coiiti'il)uted valuable researches both on frequency
curves and correlation methods. We may especially mention such
men as Guldberg, Gyllcmberg, Malmquist, Burrau and Lundquist.
In this group we might also include the Danish biologist, Johann-
^ Undersogelser over Frequensflader og Korrelation.
101 ] MODERN INVESTIGATIONS. 187
sen, whose writinfj;s on the theory of heredity are recognized as
standard texts on the apphcation of the mathematical statistical
methods to problems deahng with inherited characteristics in
organic Hfe.
A very interesting attempt to develop a theory of frequency
curves has been made by the Dutch astronomer, Kapte3'n, in his
"Skew Frequency Curves in Biology and Statistics" (Groningen,
1912). Kapteyn's theory which has much in common with
Edgeworth's method of translation introduces a new idea in the
generation of a frequency curve by making the size of the individ-
ual object depend not alone upon the sources influencing a collec-
tion of such individuals but also upon the size of the object at a
previously given time t. This idea of introducing the time factor
in the theory of probabilities is, however, more justly credited to
the French mathematician, Bachelier, whose large treatise on
probabilities of which the first volume appeared a few j-ears ago
has introduced some new thoughts regarding the conception of
continuous probabilities which are bound to strongly influence
the whole theory.
Before closing this necessarih' brief and incomplete historical
note we wish to mention the close connection of the theory of
frequenc}!- curves with that of integral equations. Since the
appearance of the epoch-making memoirs by Fredholm the theory
of integral equations has occupied a central position in math-
ematical analysis. This 3-oungest branch of higher analysis has
already found numerous practical applications in physics and
chemistry and possesses equally' important properties in the wa\' of
solving numerous statistical problems. In fact, the whole theory of
frequency curves and correlations can be reduced to the solution
of a few integral equations whose constants contain all the char-
acteristic properties of the frequency distribution. On the basis
of this principle, a complete theoiy of frequency curves could be
presented on a single book page.
CHAPTER XIV.
THE MATHEMATICAL THEORY OF FREQUENCY CURVES.
102. Frequency Distributions. — If N successive observations
originutiuf;- I'roiu the same essential circumstances or the same
source of causes are made in respect to a certain statistical variate,
X, and if the individual ol)servations o, (^ = 1, 2, 3, .... A^) are per-
muted in their natural order in accordance with their magnitude
then this particular permutation is said to form a frequency dis-
tribution of X and is denoted by the symbol F{x).
The relative frequencies of this specific permutation, that is the
ratio which each absolute frequency or group of frequencies bear
to the total number of observations, is called a relative frequency
function or probability function and is denoted by the symbol (p{x).
If the statistical variate is continuous or a graduated variate,
such as heights of soldiers, ages at death of assured lives, physical
and astronomical precision measurements, etc., then
is the probability that the variate x satisfies the following relation
z ~ -dz < X < z -\- ~dz
or that X falls between the above limits.
If the statistical variate assumes integral (discrete) values only
as for instance the number of alpha particles discharged from cer-
tain radioactive metals and gases, such as polonium and helium,
number of fin rays in fishes, or number of flower petals in plants,
then <p{z) is the probability that a: assumes the value 2. From the
above definitions it follows directly that
(a) V{z) = Nip{z) (Integral variates)
(b) dzF(z) = N(p{z)dz (Integrated variates)
Interpreting the above results graphically we find that (a) will
be represented by a series of disconnected or discn^te points while
(b) will be represented by a continuous curve.
As to the function <p(z) we make for the present no other
assumptions than those following immediately from the customary
188
103] PARAMETERS AS SYMMETRIC FUNCTIONS. 189
definition of a niatfiematical probability. That is to say the
function ip{z) iriust be real and positive. Moreover, it must also
satisfy the relation
+ 00
J (p{z)dz = 1,
— 00
or in the ease discrete variates:
2 =00
2 = -00
which is but the mathematical way of expressing the simple
hypothetical disjunctive judgment that the variate is sure to
assume some one or several values in the interval from — 0° to
+ CO. The zero point may be arbitrarily chosen and need not
coincide with the natural zero of the number scale. Thus for
instance if we in the case of Danish recruits choose the zero point
of the frequency curve at 170 centimeters an observation of
180 centimeters would be recorded as + 10 and an observation of
160 centimeters as — 10.
103. Parameters considered as Symmetric Functions, — In
regard to a frequency function we may assume a priori that it will
depend only upon the variate x and certain mathematical rela-
tions into which this variate enters with a number of constants
Xi, ^2, X3, X4 . . ., symbolically expressed by the notation
F{x, Xi, X2, X3, X 4 . . .)
where the X's are the constants and x the variate.
All these constants or parameters are naturally independent of
X and represent some peculiar properties or characteristic essen-
tials of the frequency function as expressed in the original observa-
tions Oi (f = 1, 2, 3, . . .A^). We may, therefore, say that each
constant or statistical parameter entering into the final math-
ematical form for the frequency function is a function of the
observations 0\. This fact may be expressed in the following
symbolic form
Xi = Si (Oi, O2, O3, ... Oat)
X2 = S-i (Oi, O2, O3, . . . On)
X.v = Sn (oi, 02, 03, . . . Oat)
190 THEORY OF ERRORS AND FREQUENCY CURVES. [ 103
But from purely a priori considerations we are able to tell some-
thing else about the function Si . {i = 1, 2, 3, . . . N). It is only
when permuting the various o's in an ascending magnitude accord-
ing to the natural number scale that we obtain a frequency func-
tion. This arrangement itself has, however, no influence upon
any one of the o's which were generated before this purely arbitrary
permutation took place. The ultimate and previously measured
effects of the causes as reflected in each individual numerical
observation, oi, depend only upon the origin of causes which form
the fundamental basis for the statistical object under investigation
and do not depend upon the order in which the individual o's occur
in the series of observations.
Suppose for instance that the observations occurred in the
following order
Oi, 02, 03, . . . On
By permuting these elements in their natural order we obtain the
frequency distribution F{x). But the very same distribution
could have been obtained if the observations had occurred in any
other order as for instance
07, O9, Oat, . . . O3 . . . Oi.
SO long as all of the individual o's were retained in the original
records. Or to take a concrete example as the study of the number
of policyholders according to attained ages in a life assurance
office. We write the age of each individual policyholder on a small
card. When all the ages have been written on individual cards
they may be permuted according to attained age and the resulting
series is a frequency function of the age x. We may now mix these
cards just as we mix ordinary playing cards in a game of whist,
and we get another permutation in general different from the
order in which we originally recorded the ages on the cards. But
this new permutation can equally well be used to produce the
frequency function if we are only sure to retain all the cards and
do not add any new cards.
The various functions S{oi, 02, 03, . . . on) are, therefore, sym-
metric functions, that is functions which are left unaltered by
arbitrarily permuting th(» A^ elements o, and no interchange what-
ever of the values of the various o's in those symmetric functions
can have any influence upon the final form of the frequency func-
tion or frequency curve, F{x).
] 04 ] SEMI-INVARIANTS OF THIELE, 191
We now introduce under the name of power sums a certain well-
known form of fundamental symmetrical functions defined by
the following relations
So =< + 0o + 0° + ...0^ = .V
Si = oj + 0^, + O3 + . . . On = Soi
S2 = o'l -\- oz -j- ol + . . . o§i = So?
N I N I N I N V N
sn = Oi + 02 + 03 + . . . On = 2oi
Moreover, a well-known theorem in elementary algebra tells us that
every symmetric function may be expressed as a function of
Sl, S2, S3, . . . Sn-
From this theorem it follows a fortiori that we are able to express
the constants X in the frequency curve as functions of the power
sums of the observations. Whil(> such a procedure is possible,
theoretically at least, we should, however, in most cases find it a
very tedious and laborious task in actual practice. It, therefore,
remains to be seen whether it is possible to transform these sym-
metrical functions of the power sums of the observations into some
other symmetrical functions, which are more flexible and workable
in practical computations and which can be expressed in terms of
the various values of s.
104. Semi-Invariants of Thiele. — It is the great achievement
of Thiele to have been the first mathematician to realize this
possibility and make such a transformation by introducing into
the theory of frequency curves a peculiar system of symmetrical
functions which he called semi-invariants and denoted by the
symbols Xi, X2, X3 . . .
Starting with the power sums, s», Thiele defines these by the
following identity
XiW , Xjoj' , Xaw' , „ , , „ , .■> „ , .3
Soeli !2 13 =so+— --I--2- + — ^+... (1)
which is supposed identical in respect to co.
Since s,- = So' the right hand side of the equation may also
be written as e"'" + e"^ + e"^ -f . . . = Se'^i'^.
192
THEORY OF ERRORS AND FREQUENCY CURVES.
104
Differentiating {I) witli respect to co we have
Soe
XlO) \~Cx}- /V3W'
TT"^ |2 "^ |3 ■^•
X2C0 X3W-
A, + -r|-+^- +
]
S3
So + pj-CO + 12CO- + TT^CO'^ +
X2CO X3
]■
. So 83 o , S4 3 ,
= Si + ^w + 12 w- + i^w"* +
Multiplying out and equating the various coefficients of equal
powers of co we finally have: —
Si = XiSo
52 = XiSi + X2S0
53 = XiSo + 2X2S1 + X3S0
54 = X1S3 + 3 X2S2 + 3 X3S1 + X4S0
where the coefficients follow the law of the binomial theorem.
Solving for X we have
Xi = si : So
X2 = (.S9S0 - si) : So
X3 = (S3S0 - 3s2SiSo + 2s?) : So
Xi = (.s'4.sb — 4S3S1S5 — 3s2-% + 12s2sTso — 6si) : Sq
The se:ni-invariants X in respect to an arl)itrary origui and
unit are definetl hv the relation
■%e
Xico Xow^ X.iw'
IT "^12"'^ IF "*■• •
= f"'^-\- f." ^-\- c"^"^ -\.
where Oi, 02, 03 . . . are the individual observations.
Let us now change to another coordinate system with another
unit and origin defined by the following linear transformation
Oi = aoi + c
The semi-invariants in this new system are given by the relation
Soe
TT |2 ^ 13 ■•"• •
= e'""+e"-""+ ...
= e
(aoi + r)co_j_ (ao: + c)aj
+
104] SEMI-INVARIANTS OF THIELE. 193
Since the various values of X' do not depend upon the quantity
CO we may without changing the value of the semi-invariants re-
place CO by CO : a in the above equations which give
\ CO X CO- X,CO' CO , , ^co , , ^co
a{i a-\Z a-'l-i a i "■ i "■ ,
soe = e -\- e -\- e +••..
rcoT -|
<> CO "iCO O.iCO
rco Xico X;co- Xsco'
= e Soe
Taking tlic logaiithins on both sides of the equation we have
Xi'co X/co- X;/co"^ ceo Xito Xoco"- Xjco^
'W' a^ a^ ~^^ir Tr IT
Diff(M('ntiating successivel}^ with respect to co we have
Xi' X/co Xs'co'- c Xoco X.-iCo'-
IJa a- If a' a \1 \A
\>' , X.j'co X4'co- XjCo-
X:/ Xi'cO
4+ -^ + . . . = X3+ X,'a;+. . .
a"* a'
Letting co = we therefore have
— - = — + Xi, or X/ = aXi + c
a a
— =■ = X>, or X/ = a'-'X)
a- "
X3'
— =- = X;!, or X3' = a^Xs
from which we deduce the following relations
Xi ( a.r + c) = a Xi (x) + c
Xr(a.r -\- c) = a^Xrix) for r > 1
We shall for the present leave the semi-invariants and only ask
the reader to bear in mind the above relations between X and s,
194 THEORY OF ERRORS AND FREQUENCY CURVES. [ 105
of which we shall later on make use in determining the constants
in the frequency curve (p{x).
Before discussino; the generation of the total frequency curve it
will, however, be necc^ssary to demonstrate some auxiliary math-
ematical formulae from the theory of definite integrals and integral
equations which will be of use in the following discussion as
mathematical tools with which to attack the collected statistical
data or the numerical observations.
105. The Fourier Integral Equation. — One of these tools is
found in tiie celebrated integral theorem of Fourier, which was
the first integral equation to be successfully treated. We shall in
the following demonstration adhere to the elegant and simple
solution In' M. Charlier. Charlier in his proof supposes that a
function, F(co), is defined through the following convergent series.
F{oj) = a [f(o) 4- /( a)c'^'^' +/(2a)e''^"^ + . . .
+ /(- a)^-'''^' + /(-2a)e-'«'^' + . . . ]
771 =0O
or F{co) = a^/(am)e'^"'"''
TO = - OO (2)
where i = \/- 1
We then see by a well known theorem of Cauchy that the
integral
+ 00
/(co) = ff(x)e''"dx'- ^^^
is finite and convergent. If we now let ma = x and let a =
as a limiting value, a becomes equal to dx and /(am) = fix).
Consequently we may write
Urn F((x)) = I{co).
a = o
Multiplying (2) by e~''°'^'do) and integrating between the
limits — 7r/ a and + tt / a we get on the left an expression of the
form I F(co)c~'"'^'^V/co and on the right a sum of definite integrals
-x/a
of which, however, ail but the term containing /(?• a) as a factor
will vanish. This particular term reduces to
'See Goursat: Mathetnalical Analysis (English Translation, New York),
page 364.
106] FREQUENCY FUNCTION SOLUTION OF EQUATION. 195
a I f(ra)di
lea or 2ir/(ra)
- TT a
Hence we have
+x/a
/(ra) = -^ fFioj) e-'-'^"'dco ^^^^
-•rc/a
By letting a converge toward zero and by the substitution
ra = X this equation reduces to
+ 00
fix) = ^//(co)e— -rfc- (^^^
fco
We then have, if we introduce a new function (p(o}) defined by
the simple relation :
■\/2t ^(co) = lim F{c>)), or
a=
(5a)
(5b)
•^^^^ = ^ J'^^^^^~'""^
CO
Charlier has suggested the name conjugated Fourier function of
f{x) for the expression (pi 03).
The equations (5a) and (5b) are known as integral equations of
the first kind. The expression e^"'(or e""^"') is known as the
nucleus of the equation. If in (5b) we know the value of <p{o3) we
are able to determine f{x). Inversely, if we know f{x) we may
find if^iod) from (oa).
106. Frequency Function as the Solution of an Integral Equa-
tion.^We are now in a position to make use of the semi-invariants
of Thiele, which hitlun-fo in our discussion have appeared as a
rather disconnected and alien member. On page 191 we saw
that the semi-invariants could be expressed by the relation
Xi , X2 , . Xs , .
_ V
11"- ' 12"- ' |H
e~ ~ = 2u e
when Oiii = 1, 2, 3, ... ) denotes the individual observations.
196 THEORY OF ERRORS AND FREQUENCY CURVES. [ 106
The definition of the semi-invariants does not necessitate that
all the o's must be different. If some of the o's are exactly alike it
is self-evident that the terna e"*" must be repeated as often as
Oi occurs among all of the observations. If therefore N<p(oi)
denotes the absolute frequency of Oi where (p{Oi) Is the relative
frequency function, then the definition of the semi-invariants
may be written as: —
J>j(p{Oi)e- - - = ^ ) (pjode"''^
For continuous variates, x, the above sums are transformed into
definite integrals of the form
X, , X, ,, X3 ,, +^ +°?
|1 |2 ^13 ^
e"
j (p{x)dx = I <p{x)e^"dx
Let us now substitute the quantity \/-laj, or ?'co, for co in the
above identity. We then have : —
Xi X' Xs +°° +°°
jT ^"'^ \2 '""'"^ 13 ^''^^'^ ■
e~ "
/ (p(x)dx = J <p(x)e"'"dx
under the supposition that this transformation holds in the com-
plex region in which the function is defined.
In this equation the definite integrals are of special importance.
+ 00
The factor I (p(x)dx is, of course, equal to unity according to
— oo
the simple considerations set forth on page 189. The integral
on the right hand side of the equation is, however, apart from the
constant factor '\/27r nothing more than the <p(o)) function in the
conjugate Fourier function if we let (p{x) = f{x), and
e- - - = \/27r(p(co)
According to (5b) we may, therefore* write f{x) or (p(x) as
— 00
as the most general form of the frequency function <p(x) expressed by
means of semi-invariants. (See Appendix.)
107 ] NORMAL OR LAPLACEAN PROBABILITY FUNCTION. 197
107. The Normal or Laplacean Probability Function. — The
exactness with which (p{x) is reproduced depends, of cours(% upon
the number of X's we decide to consider in the above formula.
As a first approximation we may omit all X's above the order 2
or all terms in the exponent with indices higher than 2. Bearing in
mind that r = — 1 we therefore have as a first approximation
+ 00
^o{x) = 75- / e L^ c?aj.
— <x>
This definite integral was first evaluated by Laplace by means of
the following <^legant analysis. Using the well known Eulerean
relation for complex quantities the above integral may be written as
-|-00 +°°
e 2 cos [(Xi — :r)coJ(/co+ ?" \ e ^" sin [(Xi — x)aj] rfco
— 00 — 00
The imaginary number vanishes because the factor e '^
is an even function and sin [(Xi — x)o}\ an uneven function, and
the area from — ^^ to will therefore equal the area from to
-|- 00 ^ but be opposite in sign, which reduces the total area from
— 00 to + °° or the integral in question to zero.
In regard to the first term, similar conditions hold except that
cos [(Xi — x)co] is an even function and the integral may hence
be written as
-r
+ 00
- hi 2
' COS {r(i))(I CO where r = \i — x
Regarding the parameter r as a variable and differentiating / in
respect to this variable we have
dl 2 r, . -^co^^ w ^ /
J- = — I {— hocoe - ) sm (rw) ao)
From this we have by partial integration: —
(U
dr
4-0O 00
— e - '^ sin {rco)da3 ~ ^ I e - '^ cos {rco)da)
\2 L J X2 «/
^ rT 1 dJ r
= — — or __=- —
X2 / dr X2
198 THEORY OF ERRORS AND FREQUENCY CURVES. [ 107
From \\hicli we find
log / = ~ 2V. "*" '^^ ^
where log .1 is a constant. Hence we have: —
— r-
I = Ae^'
In order to determine ^4 we let r = and we have
00
This finally gives the expression for ^„(.r) in the following form:
1 -^xr
\/27rXi
as a preliminary approximation for the frequency curve (p{x).
The first mathematical deduction of this approximate expression
for a fi-cquency curve is found in the monumental work by Laplace
on Probabilities, and the function (Po(x) entering in the expres-
sion (po{x)dx, which gives the probability that the variate will fall
between x — ^dx and x + \dx, is therefore known as the Lapla-
cean probability function or sometimes as the Normal Frequency
Curve of Laplace. The same curve was, as we have niention(>d,
also deduced independently by Gauss in connection with his studies
on the distribution of accidental errors in precision measurements.
Laplace's probability function, (po{x), possesses some remark-
able properties which it might be well worth while to consider.
Introducing a slightly different system of notation by writing
Xi = M and \/\o = a, (p„{x) reduces to the following form:
J - (x - My:2(7''
which is the form introduced by Pearson.
The frequency curve, (Po{x), is here expressed in reference to a
Cartesian (•f)()rdinate system with origin at the zero point of the
natural number system and whose unit of measurement is also
equivalent to the natural number unit. It is, however, not neces-
sary to use this system in preference to any other system. In fact,
we may choose arbitrarily any other origin and any other unit
108] hermite's polynomials. 199
standard without altering the properties of the curve. Suppose,
therefore, that we take M as the origin and a as the unit of the
system. The frequency function then reduces to
where z = (x — M) : a
Since the integral of (Poiz) from — oo to + «> equals unity
the following equation must necessarily hold.
+ 00
-t-'
This latter result may, however, be deduced independently of
the fact that <Poiz) happens to be a probability function. The
above definite integral is a form well known from the calculus and
equals ^/2T. It serves therefore as an independent check of our
calculations.
108. Hermite's Polynomials.— The Laplacean Probahility Curve
possesses, however, some otlier remarkable properties which are of
great use in expanding a function in a series. Starting with <Po{z)
we may by repeated differentiation obtain its various derivatives.
Denoting such derivatives by (pi{z), ipiiz), (fsiz) . . . respectively
we have the following relations.^
(Po[z) = e
(Piiz) = ~ z(po(z)
ip-iiz) = (z- - l)(po(z)
(pz{z) = - {z^ — ^z)(Pq{z)
(pi{z) = (z^ — 6z- + 3)(po{z)
and in general for the nth derivative: —
<Pn{Z) = (—1)2 — Z
n(n - 1) (n - 2) (n - 3)2""*
2 -4
n(n - 1) (n - 2) (n - 3) (n - 4) (w - 5)2" " ^
2-4-6
■ • <^o(
' In the followinji coiiiput.iti )iis wo have omitted temporarily the coastant
factor 1 : \/27rof (p„(z) and its derivatives.
200 THEORY OF ERRORS AND FREQUENCY CURVES. [ 109
It can be readily seen that the derivatives of <Po{z) are repre-
sented throughout as products of polynomials of z and the func-
tion iPo{,z) itself. The various polynomials
H,{z) = 1
Hxiz) = z
H.i{z) = z^ -\
H,{z) = (z' - Sz)
H,{z) = {z' - 6z' + 3)
and so forth are generally known as Hermite's polynomials from
the name of the French mathematician, Hermite, who first intro-
duced these polynomials in mathematical analysis.
The following relations can be shown to exist
H, + ,{z)+zHM + nH,,_,{z) =0
and
d'~H,Xz) zdH,Xz)
dz^ dz
nHJz) =
from which we successively may compute the various H(z).
A numerical 10 decimal place tabulation of the first six Hermite
polynomials for values of 2 up to 4 and progressing by intervals
of 0.01 is given by Jorgensen in his aforementioned 'Trekvens-
fiader og Korrelation."
109. Orthogonal Functions. — There exist now some very
important relations between the Hermite polynomials and the
derivatives of (Po{z), or between i/„(z) and <Pn(z).
Consider for the moment the two following series of functions
(po{z), (fiiz), (foiz), (p:i(z) (Pi(z), . . .
Ho{z),Ih{z),H.(z),H,(z) Ih{z),...
where (fjz) = (— 1)" H^{z) (Po(z) and where hm (p,Xz) = for 2 = ± oo
We shall now jorove that the two series <Pn(z) and //„(2) form
a biorlhogonal system in the interval — =0 to + 00 , that is to say
that they are
(1) real and continuous in the whole plane
(2) no on(> of them is identically zero in the plane
(3) every pair of th(Mn, (p^{z) and Hm{z) satisfy the relation
4-00
J (Pniz)Hm(z)dz = (n 5 w)
109 ] ORTHOGONAL FUNCTIONS. 201
We have the self-evident relation
-j-OO 4-0O -1-00
J Hm(z)(pniz)dz = J H^(z)H„{z)(Poiz)dz = J HJz)(pJz)dz
— CO — 00 — oo
Since this relation holds for all values of m and n it is only neces-
sary to prove the proposition for n > m. For if it holds for n > m
it will according to the above relation also hold for n < m.
By partial integration we have: —
-<-00 -^00 +0O
J Hm(z)<Pniz)dz = H„,{z)<p„- i{z) — J //„(2)(^„_i {z)dz
when Hm(z) is the first derivative of Hm{z).
The first member on the right reduces to since (p,^ -\{z) =0
for 2 = =t oo and because H^ is of a lower order than ^„. We
have therefore: —
4-O0 4-00
J H„Xz)<p,Xz)dz = — J H'm{z)(pn - \iz)dz
— 00 00
-f-oo -f-oo
J H'm{z)ipn-l{z)dz = — J H"„Xz)<Pn - 2{z)dz
— 00 — 00
-f-00 -t-co
J H'in(z)(Pn~2(z)dz = - J H",n{z) if^ _ i{z)dz
_ oo — 00
Continuing this process we obtain finally an expression of the form
-l-oo 4-00
jHJz)<pMdz = (- ir+\fHj"' + '^<p„^^_,{z)dz
_ oo — 00
where HJ"" ^ ^^ (z) is the m + 1 derivative of H^ (z) and
n — m — 1^0. Since H^iz) is a polynomial in the m— degree
its m 4- 1st derivative is zero, and we have finally that
4-ao
j Hm(z)<Pr,{z)dz =
— 00
for all \'alues of 7n and n where n ^ m.
202 THEORY OF ERRORS AND FREQUENCY CURVES. [ 1 10
YoY HI = n we proceed in exactly the same manner, but stop
at the in- integration. We have, ihereforc", hy replacing tn by n
in the al)ov(> partial integrations
J H^{z)iPn{z)dz = (— 1)" J Hn^''\z)<p, _ n{z)dz =
_ 00 —00
+ 00
{-\r jH^^'\z)iPo{z)dz
— 00
The n- derivative of H„(2) is however nothing but a constant and
equal to 1^. Hence we have finally
-)-oo 00 +
J H,Xz)<Pn{z)dz = (- lT\n j e-'"-^dz = (- 1)" \n^2^
-co' —00
The above analysis thus proves that the functions H^iz) and
<Pn(z) are biorthogonal to each other for all values of n different
from m throughout the whole plane.
110. The Frequency Function expressed as a Series. — We
can now make use of these relations between the infinite set of
biorthogonal functions H^iz) and (p^(z) in solving the problem
of expanding an arbitrary function (p{z) in a series of the form
(p{z) = Co<po{z) + Ci(pi{z) + 02(^2(2) + • . . ,
the series to hold in the interval from -co to + «> .
If we know that (p{z) can be developed into a series of this
form, which after multiplication by any continuous function can
be integrated term for term, then we are able to give a formal
determiiiatipn of the coefficients c.
This foi-mal determination of any one of the c's, say c„ consists
in multiplying the above series by Hi{z) and integrating each
term from — 00 to + «> . All the terms except the one containing
the product Hi{z)ip,{z) vanish and we have for c,.
j <p{z)H,{z)dz I ip{z)H,{z)dz
Ci =
ip.{z)Hi{z)dz (- l)']iV2¥
— 00
It will l)c noted that this purely formal calculation of the co-
Ill] DERIVATION OF GRAM's SERIES. 203
efficients Ci is very similar to the determination of the constants in a
Fourier Series, where as a matter of fact the system of functions
cos z, cos 2 z, cos .3 z, . . .
sin z, sin 2 s, sin 3 3, . . .
is biortho^onal in the interval ^ 2 ^ 1.
But the reader must not forget that the above representation is
only a formal one, and wc do not know if it is valid. To prove its
validity we must first show that the series is convergcMit and
secondly that it actually represents <p{z) for all values of z.
This is by no means a simple task and it cannot be done by
elementary methods. A Russian mathematician, Vera Myller-
LebedefT, has, however, given an elegant solution by means of some
well-known theorems from the Fredholm integral equations. She
has among other things proved the following criterion: —
"Every function (p{z) which together with its first two deriva-
tives is finite and continuous in the interval from — °° to + «>
and which vanishes together with its derivatives for 2 = ± <»
can be developed into an infinite series of the form: —
^{z) =y\cie-''--''H,{z)
where Hiiz) is the Hermite polynomial of order z."
111. Derivation of Gram's Series.^It is, however, not our
intention to follow up this treatment which is outside the scope of
an elementary treatise like this and shall in its place give an
approximate representation of the frequency function, (fiz), by
a method, which in many respects is similar to that introduced
by the Danish actuary Gram in his epoch-making work " Udvik-
lingsraekker," which (contains the first known systematic develop-
ment of a skew frecjuency function. Clram's problem in a some-
what modified form may l)ri(^fly be stated as follows: — Being
given an arbitrary relative frequency function, <p(z), continuous and
finite in the interval — oo ^o + °° ((^nd which vanishes for 2 = ± 0° )
to determine the constant coefficients Co, Ci, co, Cs . . . in such a way
that the series
\^iPo{z) V<Poiz) ■\/<Po(z) \/(poiz)
204 THEORY OF ERRORS AND FREQUENCY CURVES. [Ill
gives the best approximation to the quantity (p{z) : y/(po{z) in the
sense of the method of least squares. That is to say we wish to
determine the constants c in such a manner that the sum of the
squares of the differences between the function and the approx-
imate series becomes a minimum. This means that the expression
+00
r [ (p(z) ^ c,(Pi(z) -[2
must be a minimum.
On the basis of this condition we have
where the unknown coefficients c must be so determined that
+00
r r v(2) y . .
1=1 / — ^^ — C/(z) \ dzisa mmimum
y^ L V (Po(z) J
Taking the partial derivatives with respect to q we have
8c hc,J„ \^vM 8c.:/„ L J
Now since
4-00 -j-00
/ [^f/(2)Jrf2= /|c/[//„(2)J+Ci2[//i(2)J
+ . . . c„2 [^//„(2)]' I <Poiz)dz
we get
4-00 4-00
Ill ] DERIVATION OF GRAM's SERIES. 205
+00
where the latter integral equals / (pi{z)Hi{z)dz = (— 1)* K\/27r
— 00
Equating to zero and solving for c, we finally obtain the follow-
ing value for c,
+ 03
( - 1)* r
Ci = \i^2TrJ <p(z)Hi(z)dz for ^ = 1, 2, 3, . . .
This solution is gotten by the introduction of -y/lpjzj which
serves to make all terms of the form Ci(pi{z) : \^ <Po{z) equal to
■\/ <Po{z)CiHi{z) (f = 1, 2, 3, . . . n) orthogonal to each other in the
interval — «> to + oo .
In all the above expansions of a frequency series we have used
the expression <Po{,z) = e as the generating function (see
footnote on page 199), while as a matter of fact the true value of
<Po(z) is given by the equation (po(z) = e : \/27r
The definite integral on page (202)
-f-oo -|-oo
(— 1)' J Hi{z)ipi{z)dz =\ife-''-^dz = [V2^
will therefore have to be divided by \/27r, and the value of the
general coefficient c, will henceforth be reduced to
-(.00
j <p{z)Hi(z)d2
Ci =
(— l)Ni
where Hi(z) is the Hermite polynomial of order i defined by the
relation
H-(z) = i ~ iji-m-' , ^•(^• - D {i - 2) {i -^ 3) , _ 4
*^ ^ 2 2-4
- i{i - 1) (/ - 2) (^• - 3) {i ~ 4) {i - 5)z'-'^ ,
2.4.6 "^•**
20G THEORY OF ERRORS AND FREQUENCY CURVES. [ 112
On this basis we obtain tlie following values for the first four
coefficients: —
+ 00
Co = / <p{z)dz = 1
— 00
-(-00
ci ^ (- iyj(p{z)zdz:\l
— 00
+ 00
C2 = {—ir-f(z^~l)(p(z)dz:\2
— 00
+ 0O
C3 = (— lyfiz' — S2)(piz)dz : 13
— 00
+ 00
C4 = (— lyj (z' — 62- + 3z)<p{z)dz : [4
112. Absolute Frequencies. — Wliile the above development of
an arbitrary frequency distribution has reference to <p(2), or the
relative frequency function, it is, however, equally well adapted
to the representation of absolute frequencies as expressed by the
function, F(z). If A^ is the total number of individual observa-
tions, or in other words the area of the frequency curve, we evi-
dently have
+ 00 +00
N.
F(z) = Nifiz) ov j F{z)dz = N J (p{z)dz =
Since A^ is a constant quantity we may, therefore, write the
expansion of F(z) as follows:
F(z) = n\c,,<pJz) + ci(pi(z) + C2(p2{z) + . . .
e-'-'
where the coefficients Cj have the value
+ 00
Ci = ^—^ J F{z)H,{z)dz, for i = 1, 2, 3, .
112] ABSOLUTE FREQUENCIES. 207
and where
= fF(z)dz
Since all the Herniite functions are polynomials in 2 it can be
readily seen that the coefficients c may be expressed as fmictions of
the power sums or of the previously mentioned symmetrical func-
tions s, where
4.00
Sr = jz'F(z)dz
— CO
These particular integrals originally introduced by Thiele in the
development of the semi-invariants have been called by Pearson
the "moments" of the frequenc}^ function, F(z), and Sr is called
the rth moment of the variate z with respect to an arbitrary origin.
It can be readily seen that the moment of order zero or So Is
-j-OO -|-0O
So - fz''F(z)dz = N f(p{z)dz = N
— 00 — 00
Hence we have for the first coefficient Co
-f-=o 4-00
?o ^ I F{z)dz : I F{z)dz = 1
— . 03
We are, however, in a position to further simplify the expres-
sion for F{z).
As already mentioned we are at liberty to choose arbitrarily })oth
the origin and the unit of the Cartesian coordinate system for the
frequency curve without changing the properties of this curve.
Now by making a proper choice^ of this Cartesian system of refer-
ence we can make the coefficients Ci and co vanish. In order to
obtain this object the origin of the system must be so chosen that
+ 00 4-00
ci=^ f zF{z)dz :j F(z)dz =
— — 00 — 00
This means that the semi-invariant si : So = Xi must vanish.
It can be readily seen that the above expression for X: is nothing
more than the usual form for the mean value of a series of variates.
208 THEORY OF ERRORS AND FREQUENCY CURVES. [ 112
Moi-eover, we know that tho algebraic sum (or in the ease of con-
tinuous variates, the integral) of the variates around the mean
value is always equal to zero. Hence by writing for z the expres-
sion {z — M) when M equals the mean value or Xi, we can always
make C\ vanish.
To attain our second object of making Ci vanish we must choose
the unit of the coordinate system in such a way that the expression
-f-CO -|-00
ci = ^^^ J F{z)H^{z)dz : ) F{z)dz =
— 00 — CO
which implies that
-|-00 +00 -f-co
U^ F{z)zHz -J F{z)dz\ : J F{z)dz =
— 00 — oo —I — CO
or that S2 : si — 1 =0, or when expressed in terms of the semi-
invariants that
X2 = (soSo — Si") : So" = 1.
But by choosing the mean as the origin of the system the term
61 : So is equal to and we have therefore X2 = c'- = S2 : So = 1.
Hence, by selecting as the unit of our coordinate system '\/\<i
or (7, wher(> o" is technically known as the dispersion or standard
deviation of the series of variates, we can make the second coeffi-
cient Co vanish.
In respect to the coefficients C3 and c\ we have now
-fco -(-00 -f-co
(-
C3 =
73^ [J z^F{z)dz - 3 J 2F(2)rf2l: J F{z)dz
I— nn _. QD -J — CD
which reduces to — \^ — , while
4-00 4-°° +°° +°°
C4=-^|4^ J z'Fiz)dz—6J z'F{z)dz+^j F{z)dz\: j F{z)dz,
'— — 00 00 — 00 —I — CO
which reduces to
— Fl! _ ^'^2 , 3.S0I _ ± [84 _ ^"1
HLso 60 "^ soj" K U J
While the coefficients of higher order may be determined with
equal ease it will in general be found that the majority of mod-
118 J COEFFICIENTS EXPRESSED KY KF.MI-FNVARIANTS. 209
erately skow frequency distributions can be expressed by means
of the first 4 parameters or coefficients.
113. Coefficients expressed by Semi-Invariants. — We shall now
show how the same lesults for the values of the coefficients may be
obtained from the definition of the semi-invariants. Since we
have proven that a frequency function, F{z), may be expressed
by the series
F{z) = 1:c,<pXz)
we ma}^ from the definition of the semi-invariants write down the
following identity : —
XlCO X OJ^ +00
Soe = NJ e '" {Co<Po{z) + Ci<pi{z) + C2<P2(z) + . . . dz
— 00
where N is the area of the frequency curve.
The general term on the right hand side of the equation will be
of the form
+ 00
^^rfe'-iPr
{z)dz
where the integral may be evaluated by partial integration as
follows: —
+ 0O +0° +0°
J (' ''^ipr{z)dz = e'^^tpr -li?) — CO j e "^<Pr _ x{z)dz,
and where the first term on the right vanishes leaving
+ 00 +0O
J e''^<pr{z)dz = (— co)^ J e "^(^, _ i{z)dz
— 00 — 00
Continuing in the same manner we oljtain bv successive integra-
tions
+ 00 +00
(— oiYJe '"^ <pr - i(z)dz = (— coyfe "'(Pr _ 2(z)dz
— 00 — 00
+ 00 -1-00
(- ccY-fe "^ipr - 2{z)dz = (- co)^/e ^",^, _
(z)dz
210 THEORY OF ERRORS AND FREQUENCY CURVES. [114
from which we finally obtain the relation
4-00 +00 -f-oo
fe '''<Pr(z)dz = (- coffe <Poiz)dz = 77^ f e 'dz
— 00 — 00 ' — 00
V7r2
This latter integral may be written as
Consequently the relation between the semi-invariants and the
frequency function may be written as follows: —
Xiw , X.w'
Soe ~ = N \ CoCiCj + C2C0- — Csco^ + . . . e " , or
= A^ {Co — CiCO + C2CO- — C3tO^ + . . . .
-jj-+|2-(X=-l) +
Soe -
By successive differentiation with respect to co and b}' equating
the coefficients of equal powers of co we get in a manner similar to
that shown on page 192 the following results:
Co = So : N = So : So = 1
ci = — Xi
C2 = jir(Xo-i)+ xiM
C3="^| X3 + 3(X2-l)Xi = XiM
C4 = i^ [x, + 4X3X1 + 3(X2 — ly- + 6(Xo — 1) Xi^ + Xi'l
If we now again choose the origin at Xi or let Xi = and
choose y/X'z = 1 as the unit of our coordinate system we have: —
Co = 1, Ci = 0, C2 = 0, C3 = T^ X3, C4 = U Xi
114. Change of Origin and Unit. — The theoretical develop-
ment of the above formula! explicitly assumes that the variate, z,
is measured in terms of the dispersion or \/\2{z) and with Xi(z)
114] CHANGE OF ORIGIN AND UNIT. 211
as the origin of the coordinate system. In practice the observa-
tions or statistical data are, however, invariably expressed with
reference to an arbitrarily chosen origin (in the niajority of cases
the natural zci'o of the number scale) and expressed in terms of
standard units, such as centimeters, grams, ycai-s, integral num-
bers, etc.
Let us dc^note th(^ general variate in such arbitrarily selected
systems of r(^f(n'enc(; by x. Our problem then consists in trans-
forming the various semi-invariants, Xi(x'), X2(x'), X;{(.r), X^ix),
... to the system of reference with \i{z) as its origin and a/X^I^)
as its unit. Such a transformation may always be brought about
b}^ means of the linear substitution
z = a.r + b
which in a purely geometrical sense implies both a change of origin
and unit. On page 193 we proved the following general prop-
erties of the semi-invariants
Xi(2) = Xi(a.r+ b) = a\,(x) + 6
\{z) = \r(ax+ b) = a'Kix)
Let us now write Xi(.r) = .1/ and X..(.r) = a'-, we then have the
following relations: —
X,(z) = aM+b
X'iz) = <rcr'-
Since the coordinate system of r(^ferenc(^ nuist be chosen in such
a manner that \i(z) = and \/\o{z) = 1 we have
aa =1
1 ~ M . .
from which we obtain a = — and b — , which brings z on the
(7 (J
form z = [x — M) : cr, while (^,,(2) becomes
, 1 - [x-MY :2(72
Moreover, we hav(> \r(z) = \{x) : c'' for all values of r greater
than 2. We are now al)l(> to epitomize the computations of the
semi-invariants under the following simple rules :
212 THEORY OF ERRORS AND FREQUENCY CURVES. [114
(1) Compute Xi(x) ill respect to an arbitrary orighi. The
numerical value of this parameter with opposite sign is the origin
of the frequency curve.
(2) Compute X^ix) for all values of r tKjual or greater than 2.
The numerical values of those parameters divided with {Vkiix))/'
or a'', for r = 2, 3, 4, ... are the semi-invariants of the frequency
curve.
Remarks on Nomenclature and Tables. — We shall now briefly dis-
cuss some of the geometrical properties of the Laplaeean probability curve
^0(2) =e— 2-':2 and its derivatives, (pi{z) = Hi{z)(p(i(z), for i=3, 4, 5 . .
Writing ^^0(2) and its derivatives as:
(po{z)=e-z^-2:\/2TV
(Pi(z) = -zfPoiz)
(p2iz) = i-l)Hz^-l)(Po{z)
<p3iz) = {-l)Hz^-3z)(Po{z)
<Pi{z) = ( -1)4(24 -622 +3)<^o(2)
we readily notice that both ^'0(2) and all its derivatives of even order are
even functions of z while all the derivatives of uneven order are uneven
functions.
The Laplaeean probability function which occurs as a factor in all the
expressions is in itself a single valued positive function with a maximum
point at 2 =0 and a point of inflection at z = ±1 and approaches the ab-
scissa axis asymptotically in both positive and negative direction. At
2 = we have Vo(2) = l:i/27r =0.3989. At plus or minus one, <Pq{z) is less
than 0.25 at z = dz2, <Po(2) is nearly 0.05, at plus or minus 3 about 0.004
and at 2 = ±4 only 0.0001.
In regard to the third derivative, <P:i{z) =H'3(z)<^o(z) we find that it pos-
sesses a maximum or minimum in the noighliorhood of 2 = +0.7 and z =
— 0.7 respectively, it crosses the abscissa axis in the neighborhood of the
points z = zh 1.75 and approaches the abscissa asymptotically in l)oth posi-
tive and negative direction.
The fourth derivative has a major maximum point at 2 =0, it crosses the
abscisisa axis from positive to negative direction in the neighborhood of
2 = ±0.75, attains a minimum at about 2 = ±1.35, it crosses again the
ab.sci.ssa (this time from negative to positive direction) in th(^ neighbor-
hood of 2 = ±2.3, attains a secondary or minor maxinuim around 2 = ±2.86
and begins then to decline until it ultimately approaches the abscissa axis
asymptotically.
These geometrical proi)ertios of the Laplaeean frequimcy curve and its
derivatives are, however, muc^h more readily vizualized in the accom-
panying diagram which needs no further explanation. We wish, however,
to call the attention of the reader to the wavelike form of the various
curves, which is strongly reminiscent of the form of functions encountered
114]
MOMENCLATURE AND TABLES
213
in harmonic analysis or in the expansions in Fourier Series, an analogy
which we had occasion to mention in the discussion of the orthogonal
properties of the Hermite polynomials and the derivatives of the Lapla-
cean funelidn.
FIGIHE 2
In order to facilitate practical numerical calculations it is, however,
necessary to have an extensive set of numerical tables for ^o{z) and its
derivatives. This fact was already noted by Laplace who more than 25
years prior to the publication of the memoirs by Gauss on the normal
error curve advocated the construction of a table of numerical values of
the integral.
1 /»z
/27rJ
Mz*
The first set of such tables was constructed by the astronomer, Kramp
and modified forms of these tables are found in nearly all treatises on least
squares and standard texts on probabilities. The most recent set of
*This fact, as pointed out by Pearson, definitely establishes Laplace's priority of
discovery of the probability curve.
214 THKORY OF ERRORS AND FREQUENCY CURVES [114
tables of this iutofn"iil are those of Sheppard in (Tables for Bioinetricians,
edited l\v Karl Pearson) Avhere the variate z is expressed in units of <r,
or the dispersion. Sheppard has also computed a table of the numerical
values of ^0(2).
In order to use the Gram — Charlier expansion in serial form, it is, how-
ever, necessary to compute tables for the derivatives — up to the fourth
order. A brief table of the first G derivatives is already found in Thiele's
earlier treatises. Charlier was, however, the first to supply an extensive
set of tables to 4 decimal places for values of z up to 4 and progressing
by intervals of 0.01 in his Researches on the Theory of Probabilities in the
Meddelande for 1904. The most detailed tables are those of Jorgensen
in his Frekvensflnder og Korrelation which gives the values of ^0(2) audits
first G deri\atives to 7 decimal places for values of z up to 4 and progress-
ing by intervals of O.Ol.f The German astronomer Bruhns has in his
Kollektivmasslrhre given a set of tables to 4 decimal places of the values
of the definite integrals
/
(Piiz)dz for i = 0, 1, 2,3,4, 5
The Gram-Charlier series gives us the frequency function in the form
F{z) = 2ct^i(2) where the various coefficients a are expressed as moments
or semi-invariants. As we have already pointed out the derivatives of
uneven order are uneven functions and the derivatives of even order are
even functions. The addition of such terms as (-3^3(2), 05^5(2), • • • tends
therefore to produce asymmetry or skewness from the normal form, while
addition of the terras Ci<P4, c^<P(„ . . does not alter the symmetrical form
but tends to make it topheavy or flatten it around the neighborhood of the
origin or mean value of the variate 2. The coefficient cs'.S! (or X;j: 3!) is
technically known as the skewness, and ^4:4! (or X4:4!) as the excess of the
cm-ve. No particular names have as yet been proposed for the semi-
invariants of higher order.
tl intend to publish a similar sol of 5 dooimal plarc tables in tlui .socond volume of
this treatise.
PART III
PRACTICAL APPLICATIONS OF
THE THEORY
Note: — In the following pagea the factorial [^ = 1-2-3-- n is replaced
by the symbol «! and the exponent Hn"^ in the exponential expression e*""^^"^
must be interpreted as n-:2 and not as l:2n-.
CHAPTER XV.
THE NUMERICAL DETERMINATION OF THE PARAMETERS
115. General Remarks. — The previous investigations on
frequency functions have all been more or less of a purely
theoretical nature. In the present chapter we now propose
to show how the parameters are determined in actual practice
from the individual observations or statistical summaries
The determination of these unknown co-efficients or para-
meters can^ — as emphasized by J0rgensen in his Frekvensflader
og Korrelation — be looked at from two points of view. We
may either consider the series as infinite in which case the ques-
tion of determining the co-efficients becomes a problem in the
Theory of Functions; or we may decide to consider a finite
number of terms in the series and determine the coefficients so
that the sum of the squares of the deviations of the resulting
function from the observed statistical data becomes a minimum
in the sense of the method of least squares. In this case the
coefficients and not the moments or semi-invariants are repre-
sentative of the observations. This latter method is the classi-
cal method as used by Gram in his fundamental research on
the expansion of frequency functions in series. A brief state-
ment of the essential differences of the two methods may, how-
ever, be of advantage to the reader.
The method of moments requires that the areas of the defi-
./■
nite integrals of the form I x''F{x)dx must equal the areas
— X
of the observations which are expressed as power sums of the
form
X= M
' T.^'F{x)
X= — =0
while the method of least squares requires that
J [F{x)-Y.c.^.ix)Ydx
215
216 DETERMINATION OF PARAMETERS [116
must equal a minimum but does not necessarily impose any re-
strictions as to the condition of equality of the observed and the
computed areas as derived from the mathematical formula.
The pi'oblem of determining the parameters in the sense of
the method of least squares is therefore essentially a simple
problem in maximum and minimum and is not necessarily —
as some critics have imagined it to be — invariably interwoven
with the law of errors as expressed by the Laplacean probability
curve. It is, of course, true that the law of errors can be proved
by regarding the principles of the method of least squares as
an axiom, and inversely by accepting the law of errors as an
axiom, i. e. by assuming the deviations from the observations
and the functional law or mathematically determined frequency
curve as being due to chance or random sampling, we may
prove that the sum of the squares of such deviations actually
becomes a minimum. This peculiar relation is, however, not
necessary or required when we view the determination of the
constants as a simple problem in maxima and minima.
116. Remarks on Certain Criticisms. — Many English and American
actuaries ha\ e of late shown a tendency to ignore the method of least
squares and i)refer to rely entirely upon the method of moments. Thus
Palin Ehlerton in his otherwise usefid and instructive wcn'k onl'rcquency
Curves and Correlation states that the method is of little practical use,
while ]Mr. D. Caradog Jones in his newly ]>ul)lished /'"(7-.s/ Course in Statis-
tics claims the method of least scjuan's "wliich is the traditional way of
approaching all such problems, is shown to be impracticable in a large
number of cases, either because the resulting equations cannot be solved,
or, when they are capable of solution, because the labour involved would
be colossal." This objection falls, however, to the ground in the case
of the expansion of a frequency function in serial form because the un-
known parameters, with the exception of the origin (the mean) and the
unit (tlu; dispersion) of the co-ordinate system, all appear as coefficients
in true liruuir equations and hence are eminently adaptable to the treat-
ment by least s(iuarcs.
The attitude of these writers is probably due to the fact that they work
exclusively with the Pearsonian type of frexjuency curves where the
function, F(z), is given as a closed expression rather than as an expansion
in serial form. In nearly all IVarson's cur\e types there appear not more
than four constants wliich in a measure acciounts for the often successful
application of the method of monu^nts, although several of the examples
presented })%• Mr. .Jones in his book can scarcely lie said to be recommenda-
tive to Pearson's theory. On the other hand, it is a great drawback, not
being al)le 1o have mor(> than four constants at our disposal. Personally
I have encountered a large number of statistical s<'rii'S where the Pear-
sonian theory fails. This same fact is also noted l)y Jorgensen who on
Fiage -V.) of his l-'rek-rensjUnler og Korrelation states that "jeg kender flere
agttagelsesraekker. livor Pear.son's Teori svigter lot alt."
In the purely theoretical development it matters but little whether we
use moments or least squares in the expansion of a frecpiency function in
a series; a fact which is readily seen from our previous d(!monstrations.
In the purely practical work wo have, however, this fact to consider,
116] REMARKS ON CRITICISMS 217
that the method of moments works exelusiv^ely with ansas expressed as
definite integrals, which are often ditlicult to deti-rmine in extremely
skew distributions. And it is only by successive approximations that we
in this manner reach a i)hiusil)le result. Moreover, unless the observa-
tions are very numerous, it is almost hopeless to comi)ute tlui moments
of higher order than the fourth, because of the very largt; (irrors arising
from random sanii)liug. ("iiarlier in one of liis monographs asserts that
it is generally ust>lt'ss to comi)ute moments of higher order tiian the
second when the number of individual ol)servations in the statistical
series is less than 1000. Thiele gives the following brief rules:
For the first and second semi-invariants rely exclusively on the observed
data.
For semi-invariants of higher order than 6 rely exclusively on theoretical
considerations.
For intermediate semi-invariants (between the 2d and 6th) rely partly
upon theory and partly upon the observations.
Caradog Jones, on the other hand, lustily ventures forth with moments
of the fourth order, based upon 241, and in some instances even as low as
180 indi\idual observations. It is, therefore, no wonder that some of his
results exhibit a somewhat poor "fit" with the original data. Another
criticism which may be lodged against the method of moments as used
by some adherents of the Pearsonian school, rather than l)y Pearson him-
self, is that it works with unweighted observations, and the values of the
extremities of the frequency cur\es are gi\'en the same weights as the
more numerous observations in the immediate neighborhood of the mean.
A second objection, raised among others by Elderton, is that the ex-
pansion in serial form sometimes gives rise to negative frequencies at the
extreme tail ends of the curve, due of course to the fact that we have used
a limited number of terms of the series. From purely practical con-
siderations this objection counts little, because the observations at the
extremeties are very few in numlier. It matters, for instance, but little
in ordinary calculations of assurance premiums whether the upper limit
of a mortality table is at 90 or at 100, and when Pearson from liis curves
actiudly has attemi)ted to put an ui>pfr limit to the duration of human
life, he has, to l)orrow an expression from the Danish biologist, Johannsen,
begun to handle biology (is mathematics and not with mathematics. In
this connection it may also be noted that the Pearson Type I curve gives
imaginary values beyond certain limits. When now certain followers of
the Pearsonian school have considered this as an advantage and tried to
interprete the limits as possbile values of repeated or i)resumpti\e observa-
tions, it seems that such disciples have stretched their poinlJ a bit too far.
It is not possible to see why negative results should be less plausible than
imaginary results. Every student of ordinary algebra knows that the
"iniaginary" quantities are just as valid as the so-called "real" quantities,
and it is prol>ably the choice of this unhai>py and ill-chosen nomenclature
which has gi\en rise to the abo\e extravagant claims of some of the fol-
lowers of Pearson.
Finally some English and American actuaries have objected to the
arbitrary choice of the parameters in the Oram or Charlier expansions.
Unless i have completely misunderstood Mr. Elderton this is one of his
his chief criticisms against Charlier's method. With my best intentions
I cannot agree to this and will even go so far as to say that Mr. Elderton's
criticism really speaks in favor of the methods put'forth by the Scandi-
navian scholars. As we have repeat I'dly emphasized in the preceding
paragraphs, the arbitrary choice of ci and c-2 amounts mathematically to
the choice of an arbitrary origin and unit in the Cartesian co-ordinate
system to which surely ru) mathematician will make objections. Neither
can objections be raised from t lu^ point of view of common sense. We
might as well object to the meter as a uiut of measure in preference to the
yard, or to reckoning the solar time from the Greenwich meridian instead
of the meridian of Paris.
218 DETERMINATION OF PARAMETERS [117
The failur(> of tlu' luctliod ol' iiioiiuMils to coTiiputc with any degree of
aceunK'v luoineiits of liigher order in tlie case of the niajority of ordinary
obser\ations is i>robably the reason why some actuaries, especially in
America, have maintained that the Gram or Cliarlier A tyi>e of fre-
quency curAes is not ])owerful enough to represent more than moderately
skew i'requt'ncy distributions.
In spite of the incontrovertible fact that the most recent researches in
the theory of integral equations have demonstrated beyond doubt that
any freqiu'ucy curve can be de\("loped in convergent series by Hermite
polynomials in conjunction with tlie normal Laplacean frequency curve
an American actuary, Mr. Merwyn Davis, has taken the "bull by the
horns" so to speak and lioldly gone on record with the positive statement
that "the C'harlier series fails completely in cases of appreciable skewness."
With all due respect for this young matador who has so boldly entered
the ring to challenge the work of some of the most eminent mathemati-
cians in th(^ realm of integral equations I feel, however, that if Mr. Davis
has actually succeeded in "throwing the bull" it is only in the sense as
implied in the colloquial slang of his native America. In fact, we shall
presi'utly in some of our examples take up the challenge of Mr. Davis
and show that the series he so curtly rejects can — by means of a simple
transformation — be ixsed on decidedly skew frequency distributions with
even greater success than the Pearsonian curve types.
With these preliminary remarks we shall now proceed to
give several examples of the application of the Gram or Lapla-
cean — Charlier frequency series, employing either the method
of moments or the method of least squares in the numerical
determination of the constants, although preference will be
given to the latter method in cases of appreciable skewness or
excess.
It is, however, not our intention to go into details of the
method of least squares and its relation to error laws, except
in its connection with the problem of maximum and minimum.
Any number of standard treatises are now available on the sub-
ject, however, to which we may refer interested readers.*
117. Charlier's Scheme of Computations. — The general
formulae for the semi-invariants were given on page (192).
In practical work it is, however, of importance to proceed along
systematic lines and to furnish an automatic check for the cor-
rectness of the computations. Several systems facilitating such
work have been proposed by various writers but the most
simple and elegant is probably the one proposed by M. Charlier
and which is shown in detail with the necessary control checks
on the following pages. Charlier employs moments, while we
in the following demonstration shall prefer the use of the semi-
invariants.
* A particularly attractive presentation in Englisii is found in David Brunt's Com-
bination oj ObHrrvations (Cambridge, 1918).
117] charlier's computation scheme 219
If we define the power sums of the relative frequencies ^{x)
f-\- .o r'^ CO
x'F{x)dx: I F{x)dx for r = 0,1,2,3 . . , .
we find that the expressions for the semi-invariants as given on
page (192) may be written as follows:
Xi =mi
/l2 = TO2— mr
/L-! = ma — Sw-iWi +2mi'''
X4 = W4 — 4???.tmi — ^m-r + 12?W2?Wi" — 6wi^
The advantages of the Charlier scheme for the compuation
of the semi-invariants lies in the fact that it furnishes an auto-
matic check of the final results. If we expand the expression
{x-\-iyF{x) we have:
x'F{x) +4x^F(x) +6x-T(x) +4xF(x) -\-F{x) or
2](x + l)T(a:)=S4+4s3-f6s2+4si+So,
which serves as an independent control check of the computa-
tions. Moreover, another check is furnished by the relation
W4 = X +4miX3 + 6mi-X2 + 3Xo- + mV.
In order to illustrate the scheme we chose the following age
distribution of 1130 pensioned functionaries in a large American
Public Utility corporation.
Ages
No. of Pensioners
Ages
No. of Pensioners
35-:i9
1
6.5-69
28j
40-44
6
70-74
248
45-49
17
7r>-79
128
50-54
48
80-84
38
55-59
118
8.5-89
13 .
60-64
224
o\'(T 90
3
Choosing the age of 67 as a provisional origin the Charlier
scheme is shown in detail on next page.
The computation below gives the numerical values of the
frequency function which now may may be written as follows:
F(x)=1130[<^oU)+.0258.^3(x)+.0158^4(x)]
where -[Ct.^^Y
(f(i(x) = e
1.624 \ 27r
220
DETERMINATION OF PARAMETERS
[118
Ages
35-39
40-44
45-49
50-54
55-59
60-66
65-69
F(x)
1
6
17
48
118
224
2S6
700
xF{x)
6
30
68
144
236
224
a;2F(x)
36
150
272
432
472
224
x3F{x)
216
750
1,088
1,296
944
224
x*F(x) (z+l)*F(z)
708
1,586
1,296
3,750
4,352
3,888
1,888
244
4,518 15,418
625
1,536
1,377
768
118
286
4,710
70-74
+ 1
248
248
248
248
248
3,968
75-79
+2
128
256
512
1,024
2,048
10,368
80-84
+3
38
114
342
1,026
3,078
9,728
85-89
+4
13
52
208
832
3,328
8,125
90-94
+5
2
10
50
250
1,250
2,592
over 95
+6
1
6
36
216
1,296
2,401
2 430
686
1,396
3,596
11,248
37,182
Sr 1,130
— 22
2,982
-922
26,666
41,892
wir= 1.0000 -.0195 2.6378 -.8156 23.5699
Xi =mi = —
.0195
m2= 2.6378
S4= 26,646
Xi2=wi2 =
.0004
-mi2 = -.0004
4s3 = -3,688
Xi^ = mi^ =
.0000
X2= 2.6374=0-2
6s2= 17,892
Xi^ =rai'* =
.0000
■Vxi= 1.6240=0"
4si = - 88
4.2831=0-3
«o= 1,130
6.9558=0-4
41,892
wi2mi = —.05
13, i/ismi
. = .0159 m-i^ =6.9580, moniy^ = .0010
mz = -
-.8156
nn = 23.5699
X4= 2.6450
— 3m2TOi =
.1539
— 4»i3Wi = — .0636
4miX3 = .0516
2mi3 =
.0000
-3^22 = -20.8740
6mi2X2 = 0060
X3=-
.0017
12m2TOi2 = .0127
3X22=20.8677
-6wu4 = .0000
X4 = 2.6450
wu" = .0000
23.5703 =W4
C3 = X3 -.a^ = -
-.1545
C4=X4:o-4=.3803
-C3:3!=.0258
C4:4! = +.0158
118. Comparison Between Observed Data and Theo-
retical \ allies. — The next step is now to work out the numeri-
cal values of F(x) for various values of x and compare such
values with the ones originally observed. This process is shown
in detail in the following scheme:
119 J OBSERVED AND THEORETICAL VALUES 221
(1) (2) (3) (4) (o) (Oj (7) (8) (9) (10) Obs.
X I— Xi {x—\^)■.a^p^^{^) ifsiz) (p^iz)
-7 -G.9S -4.:«)0 .OOOl +.0058 +.0170 +.0001 +.000:5 .000.')
-6 r.<)s :5.()S'j .000.-) .0170 .0470 .001.') .ooos .oo-js ■_' 1
-.5 4 OS :5.0()7 .()():)() .0710 .12()7 .OOIS .OOiO .0074 .5
-4 3 OS 2.4.-)l .019S .14.-)S +.0()()2 .0(«S +.0000 .024.-) 17 17
-3 2.98 1.835 .0741 +.0.500 -.4345 +.0013 -.00()S 0()S() 48 48
-2 1.98 1.219 .1897 -.3.502 -.7036 -.0090 .01 1 1 . KiOO 118 118
-1 -0.98 -0.()03 .3320 -.5287 +.31()0 -.013() -.00.-)0 3140 219 224
+0.02 +0.012 .3989 +.0143 1.1963 +.0004 +.0189 .4182 291 286
+ 1 1.02 0.628.3273 ..53.59 +.2.584 .0138 +.0041 .34.52 241 248
+2 2.02 1.244 .1835 +.3325 -.7157 +.0086 -.0113 .1808 126 128
+3 3.02 1.8()0 .0707 -.0605 -.4094 -.0015 -.()()()5 .0()27 44 38
+4 4.02 2.475.0186 .1443 +.0703 .0037 +.001 1 .0212 15 13
+5 5.02 3.091 .0034 .(UiSO .1241 .0018 .0020 .003() 3 2
+6 6.02 3.707 .0004 .0165 .045() .0004 .0007 .0007 1 1
+7 +7.02 +4.322 .0001 -.0050 .0162 -.0001 +.0003 .0003
Column (1) gives the values of the variate x reckoned from
the provisional origin, or the centre of the age interval 65-69.
(2) is X less the first semi-invariant, whereby the origin is shifted
to the mean or a. Column (3) represents the final linear trans-
formation : z= {x — Ai) :<T.
Columns (4), (5) and (6) are copied directly from the stand-
ard tables of J0rgensen or Charlier. Column (7) is (5) multiplied
by 0.0258 or the product - [csiPsiz) ] :3 !, while (8) is [d^^iz) J :4 !.
Column (9) is the sum of (4), (7) and (8) If we now distri-
bute the area N = so or 1130 PRO rata according to (9), we
finally reach the theoretical frequency distribution expressed
in 5-year age intervals and shown in column (10) alongside
which we have inserted the originally observed values. Evi-
dently the fit is satisfactory. It will be noted that the final
frequency series is expressed in units of 5-year age intervals.
This, however, is only a formal representation. By subdividing
the unit intervals of column (1) in 5 equal parts, and by com-
puting all the other columns accordingly, we get the theoretical
frequency series expressed in single year age intervals.
119. The Principle of Method of Least Squares.— The
following paragraph purports to give a brief exposition of the
determination of the coefficients in the Gram or Laplacean —
Charlier series in the sense of the method of least squares as a
strict problem of maxima and minima, wholly independent of
the connection between the method of least squares and the
error laws of precision measurements.*
*In the following demonstration I am adhering to the brief and lucid exposition
of the Argcmineau actuary, U. Broggi, in his e.xellent Traite d' Assurances sur la Vie.
222 DETERMINATION OF PARAMETERS [119
The simple problem in maxima and minima which forms the
fundamental basis of the method of least squares is the follow-
ing: Let VI unknown quantities be determined by observations
in such a manner that they are not observed directly but enter
into certain A- /^o;/'/i functional relations, /,(.ri, 0:2, x.i, .... x„,),
containing the unknown independent variables, X\, X2, xs, . . . x,„.
Let furthei-more the number of observations on such functional
relations be ii (where n is greater than w). The problem is
then to determine the most plausible system of the values of
the unknown x's from the observed system.
/l (Xi, X2, x^, . .
. . Xj=0i
/2 (Xl, X2, X3, . .
. . Xj =02
/„ (Xi, X2, Xz, .
• • X „J 0„
when /i, j-i, . . . /„ are the known functional relations and
oi, 0-2, . . . o„ their observed values. Such equations are known
as observation equations.
In order to further simplify our problem we shall also assume
that
1 All the equations of the system have the same weight, and
2 All the equations are reduced to linear form.
By these assumptions the problem is reduced to find m in-
knowns from n linear equations.
ai xi
+ 6.
X2
+ . .
. =01
ai x\
+ 62
Xi
+ . .
. =02
03X1
+ 63
Xi
+ . .
. =03
a„ xi
1 + 6,
, X.
I + .
. . =0,
Since n is greater than m we find the problem over-deter-
mined, and we therefore seek to determine the unknown quan-
tites, Xi, Xj, . .x,„ in such a way that the sum of the squares
of the differences between the functional relations and the ob-
served values, becomes a minimum. This implies that the
expression
i = n
X(a.a:i+6.X2+ . . . -o,)' = i/'(xi, X2, . . . xj
i = l
119j METHOD OF LEAST SQUARES 223
must be a minimum or the simultaneous existence of the equa
tions.
dx\ ' dx2 ' dx ,n
If we now introduce the following notation
aiX\ + h,x-2-{- . . . — 0, = "a, for ^ = li 2i 3, . . . n,
The w equations in the above system (/) evidently take on
the following form
Aiai + AL'a-j+ . . . +A„a„=0
■A161 + X2&2+ . . . +-A„6„=0
If we now again re-substitute the expressions for "a in terms of
the linear relations
aiXi^b,X2+ . . . 0, = A„ for ?' = 1, 2, 3, . . .n,
and collect the coefficients of Xi, x^, . . . x„, these equations
may be expressed in the following symbolical form:
[aa]xi + [ab]x2 +....— [ao] =
[ab]xi + [bb]x2 +....- [bo] =
[ak]xi + [6^-].r2 +....+ [kkU,„ - [ko] =
where [aaj = Oi'- + ar + . . . .
[ab\ = Oi 61 + 02 1'2 + . . . .
is the Gaussian notation for the homogenous sum products.
The above equations are known as normal equations, and it is
readily seen that there is one normal equation corresponding
to each unknown. Our problem is therefore reduced to the
solution of a system of simultaneous linear equations of m un-
knowns. If m is a small number, or, what amounts to the same
thing, there are only two or three unknowns the solution can be
carried on by simple algebraic methods or determinants. If
the number of unknowns is large these methods become very
laborious and impractical. It is one of the achievements of the
great German mathematician, Gauss, to have given us a
method of solution which reduces this labor to a minimum and
224 DETERMINATION OF PARAMETERS [120
which proceeds along well defined systematic and practical
lines. The method is known as the Gaussian algorithmus of
successive elimination.
120. Gauss' Solution of Normal Equations. — For the sake
of simplicity we shall limit ourselves to a system of four normal
equations of the form
[aa]xi + [ab]x2 + [ac]x3 + [ad]xi — [ao] =
[ab]xi + [bb]xi + [bc]x3 + [bd]xi — [bo] =
[ac]xi + [bc]x2 + [cc]x3 + [cd]Xi — [co] =
[ad]xi + [bd]x2 + [cd\x3 + [dd]xi — [do] =
The generalization to an arbitrary number of unknowns offers
no difficulties, however.
On account of their symmetrical form the above equations
may also be written in the more convenient form, viz. :
[aa]xi + [ab]x2 + [acxs + [ad]xi — [ao] =
[bb]x2 + [bc]x3 + [bd]Xi - [bo] =
[cc]x3 + [cd]Xi — [co] =
[dd]Xi - [do] =
From the first equation we find
[ao] [ab] [ac] [ad]
Xl = . - - X2 - - X3 - ~-Xi
[aa] [aa] [aa] [aa]
Substituting this value in the following equations and by the
introduction of the new symbol
[ik] — , ' [ak] = [ik.l]
[aa]
we now obtain a new system of equations of a lower order and
of the form
[bb.l]x2 + [bc.l]x3 + [bd.l]xi - [bo.l] =
[c'c.l]x3 + [cd.l]xi — [co.l] =
[ddA]x4 - [do.l] =
Solving for X2 we have
^ [bo.l] _ [6c. 1 ] _ [bd.l]
''' [66.1J [66.1]'^' " [66.1]'^'
121] GAUSS' SOLUTION 225
Substituting in the following equations and writing
166.1]
we have
[cc.2]x3 + [cd.2]x4 = [co.2]
[dd.2]xi = [do.2], or
_ [co.2] _ [crf^l
"'^ " [CC.2] [cc.2f'
Moreover, by writing
\ck 21
[ik.2] - [ci.2\ "^^ = [ik.S]., we have finally
[cc.2]
[dd.S]Xi = [do.Z]
This gives us the final reduced normal equation of the lowest
order. By successive substitution we therefore have:
X4 =
[do.S]
[dd.S]
_ [co.2] _ [cd.2]
""' ~ [cc.2] ~ [cc.2]''"
_ [ho.l] _ [6cll _ [6d.l]
^' ~ [66.1] [66.1]^' [66.1]^'
[ao] [ab] [ac] [ad]
Xi ^ . , - , -X2 X3 - Xi
[aa] [aa] [aa] [aa\
as the ultimate solution of the unknowns.
121. Arithmetical Application of Method. — The example
in paragraph 117 gave an illustration of the application of the
method of moments. As previously stated this method works
quite well in cases of moderate skewness, but is less successful
in extremely skew curves and where the excess is large. We
shall now give an illustration of the calculation of the para-
meters by the method of least squares. The example we choose
is the well-known statistical series by the distinguished Dutch
226 DETERMINATION OF PARAMETERS [121
botanist, deVries, on the number of petal flowers in Ranunculus
Bulbosus.* This is also one of the classical examples of Karl
Pearson in his celebrated original memoirs on skew variation.
Although the observations of deVries lend themselves more
readily to the method of logarithmic transformation, which we
shall discuss in a following chapter, we have deliberately chosen
to use it here for two specific reasons Firstly it is a most strik-
ing illustration in refutation of the incautious criticism of the
Gram-Charlier series by the aforementioned Mr. Davis. Sec-
ondly (and this is the more important reason) it offers an ex-
cellent drill for the student in the practical applications of the
method of least squares because it gives in a very brief compass
all the essential arithmetical details. The observations of
deVries are as follows :
i. of Petals
X
F{x) =0.
5
133
6
1
55
7
2
23
8
3
7
9
4
2
10
5
2
where F(x) denotes the absolute frequencies. The observed
frequency distribution is well nigh as skew as it can be and rep-
resents in fact a one-sided curve, and should therefore — if the
statement by Mr. Davis is correct — show an absolute defiance
to a graduation by the Gram-Charlier series.
The process we shall use in the attempted mathematical
representation of the above series is a combination of the method
of semi-invariants and the method of least squares. Following
Thiele's advice we determine the first two semi-invariants in
the generating function directly fr-om the observations while
the coefficients of this function and its derivatives are deter-
mined by the least square method.
Choosing the provisional origin at 5, we obtain the following
values f(jr the crude moments.
*rnf|<)ulii<(lly many nwlirs will think that I have spent an luuisnal lii-avy amount
of arithmiiii- on such a simple (■xampli\ This criticism is true, and in actual curve
fitting practice we would of course re-sort to a logarithmic transformation. The ex-
ami)le is in this particular instance chosen as a drill for I lie si iident . I may. however
remark that if one weri? to us(? I'ersonian curves the arithmetical work would bo even
more formidable than through the application of least squares: because we would
have to rt^sort to mechanical quadrature formulas in order to compute the areas in
Pearson's curves.
121] APPLICATION OF METHOD 227
So =^222, Si-140, Si =292, S3 = 806, S4 =2,752, S5 = 10,790, S6 =
46,072, si =207,226, from which we find that
Xo = l, X, =0.631, A2 = 0.917, 7.3=1.644, A4 = 3.377, Ar. = 5.972,
X6= -2.911, AT =-122.638.
All these semi-invariants with the exception of the two
first are howevei', so greatly influenced by random sampling
in the small observation series that it is hopeless to use them in
the determination of the constants in the Gram-Charlier series.
In fact an actual calculation does not give a very good result
beyond that of a first rough approximation. The generating
function, on the other hand, may be expressed by the aid of
the two first semi-invariants as follows:
I -z2:2
where z is given by the linear transformation:
z = (:r - 0.631 ):0. 9576. (v">^> = 0.9576).
We now propose to express the observed function F{x) or
(p{z) by a Gram-Charlier series of the form:
F{x) = <p{z)=^koiPQ(z)-\-k-i^i{z)+kA^i{z)-{- h(pi{z).
In this equation we know the values of the generating func-
tion and its derivatives for various values of the variate z as
found in the tables of J0rgensen and Charlier, while the quanti-
ties k are unknowns. On the other hand we know 6 specific
values of F{x) as directly observed in deVries's observation
series. We are thus dealing with a system of typical linear
observation equations of the forms described in paragraphs
119 and 120 and which lend themselves so admirably to the
treatment by the method of least squares.
From the above linear relation between x and z we can directly
compute the following table for the transformed variate z.
-0.688
1
+0.402
2
+ 1.493
3
+2.583
4
+3.674
5
+4.764
228 DETERMINATION OF PARAMETERS [121
The numerical values of s^u(2) and its derivatives as corres-
ponding to the above values of z can be taken directly from the
standard tables of J0rgensen and Charlier. We may there-
fore write down the following observation equations (a)
S^O 9z ^4 ^5 ^6
.314SAo
-.5472A-3
+ .1207A-4
+ 2.272SA-5
+ .9591A-6
-133
=
.8()79Ao
+.419SAs
+ .7566A-4
-1.9S36A-5
-2.9860A-6
- 55
=
.l.SOSAo
+ .150f.A-3
-.7073A-4
+ .4540A-5
+2.8600A-6
- 23
=
.014r,A-o
-.134GA:,
+ .l()02A-4
+ .2()42A-5
-1.21H()A-6
- 7
=
.OOOoAo
-.OlSOA-3
+ .04S(iA-4.
- .1070A-5
+ .14,S2A-6
- 2
=
.OOOUo
-.0005A-3
+ .0020A.4,
- .0()4;U-5
+ .054()A-6
_ 2
=
for which we now propose to determine the unknown values of
k by the least square method.
While this method may of course be applied directly to the
above data, it will generally be found of advantage to start
with some approximate values of the /c's. It is found in prac-
tice that this approximate step saves considerable labour in
the formation and ultimate solution of the normal equations.
Although the first approximation in the case of numerous un-
knowns must be in the nature of a more or less shrewd guess,
which facility can only be attained by constant practice in rou-
tine mathematical computing, we are, however, in this specific
instance able to tell something about the nature of the coeffi-
cients from purely a priori considerations. W^e know for in-
stance from the form of the Gram-Charlier series that the
coefficient ko of the generating function must be nearly equal
to the area of the curve, which in this particular instance is 222.
Moreover, a mere glance at the observed series tells us that it
has a decidedly large skewness in negative direction from the
mean coupled with a tendency of being "top heavy," indicating
positive excess. We can therefore assume as a first approxima-
tion that the coefficients of the derivatives of uneven order are
negative and the coefficients of derivatives of even order are
positive. Again, it is also seen that the coefficients of the de-
rivatives of higher order than the fourth must be relatively
small in comparison with the coefficients of the derivatives of
lower order, otherwise the series would not be rapidly con-
vergent.
From such purely common sense a priori considerations we
therefore guess the following first approximations, viz.:
A;'o = 222, A;'3=-10(), k\ = 20, k''o=-5, k\ = 5.
121] APPLICATION OF METHOD 229
The probable values of the various k's may be written as
ki = r,k\ for i = 0, 3, 4, 5 and 6,
and our problem is therefore to find the correction factor r, with
which the approximate value k\ must be multiplied so as to
give k,.
Applying the various values of k\ to the original observation
equations in (a) we obtain the following schedule for the numeri-
cal factors of r,.
a
b
c
d
e
j
699
+547
+ 24
-144
+ 48
-1,330
-126
817
-420
+ 151
+ 99
-149
- 550
- 52
290
-151
-141
- 23
+143
- 230
-112
32
+ 135
+ 21
- 13
- 61
- 70
+ 44
1
+ IS
+ 10
+ 5
+ 7
- 30
+ 11
+
+
+
+ 3
- 10
- 7
1839 +129 +65 - 46 - 9 -2,220 —242
where the additional control column s serves as a check.
The subsequent formation of the various sum-products and
normal equations is shown in the following schedules together
with the s columns as a check.
aa
ab
ac
ad
ae
ao
as
488,601
+382,353
+
16,776
-
79,686
+ 33,552
-
929,670
—
88,074
667,489
-343,140
+ 123,367
+
80.SS3
-121,733
—
449,350
—
42,484
84,100
- 43,790
-
40,890
-
6,670
+ 41,470
-
66,700
-
32,480
1,024
+ 4,320
+
672
-
416
- 1,952
-
2,240
+
1,408
1
+ 18
+
10
+
5
+ 7
-
30
+
11
+
+
+
+
-
1,241,215
239
+
99,935
-
5,884
- 48,656
-1,447,990
-161,619
66
6c
bd
be
bo
6s
299,209
+
13,128
-
63,258
+ 26,256
-
727,510
—
68,922
176,400
-
63,420
-
41,580
+ 62,580
+
231,000
+
21,840
22,801
+
21,291
+
3,473
- 21,593
+
34,730
+
16,912
18,225
+
2,835
-
1,755
- 8,235
-
9,450
+
5,940
324
+
180
+
90
+ 126
-
540
+
198
+
+
+
-
_+_
516,959
—
25,986
-102,130
+ 59,134
-
471,770
-
24,032
cc
cd
ce
CO
cs
576
-
2,736
+ 1,152
-
31,920
_
3.024
22,801
+
14,949
- 22,499
-
83,050
-
7,852
19,881
+
3,243
- 20,163
+
32.430
+
15.792
441
-
273
- 1,281
—
1,470
+
924
100
+
50
+ 70
-
300
-L
110
+
+
-
-
43,799 + 15,233 - 42,721 - 84,310 + 5,950
230
DETERMINATION OF PARAMETERS
[121
dd
12.996
9,801
529
169
25
de
5,472
14,751
3.289
793
35
■2 +
do
151.620
54,450
5,290
910
150
ds
+ 14,364
- 5,148
+ 2.576
572
+ 55
+
23,520
- 22,684
+
103,220
+
11,275
ee
eo
es
2,304
-
63,840
-
6,048
22,201
+
81.9.50
+
7,748
20.449
-
32.890
-
16.016
3,721
+
4,270
-
2.684
49
-
210
+
77
9
-
30
-
21
48,733
10,750 - 16,944
We may now write the normal equations in schedule form as
follows :
(1)
(b)
(2)
(c)
(3)
id)
(4)
(e)
Original Normal Equations
(a) 1,241,215 - 239 + 99935
5884
48656 -1447990
+
-
19
+
1 +
9 +
278
+516959
- 25986
-102130
+ 59134
-
471770
+ 8046
- 47 4
- 3917
-
116582
+ 43799
+ 15233
- 42721
-
84310
+ 28
+ 231
+
6865
+ 23520
- 22684
+
103220
+ 1907
+
56761
-f 48733
-
10750
(o)
.00019
+ .08051 -.00474 -.03920 -1.16659
The sum-products from the observation equations are shown
in the rows marked (a), (6), (c), (d) and (e). The row marked
(5) and printed in italics is formed by dividing each of th
figures in row (a) with 1,241,215. The row marked (1) con-
tains the products of the figures in row [a) multiplied with the
factor .00019. Row (2) is the products of the factor 0.08051
and the figures in row (a), while row (3) is the product of the
factor — 0.00474 and the figures in row (a). The products in
row (4) are formed in the same manner by means of the factor
-0 03920.
We next subtract row (1) from row (b), row (2) from row (c),
row (3) from row (d), and so forth, which results in the follow-
ing schedule, which is known as the first reduction equation.
121]
APPLICATION OF METHOD
231
(a)
(1)
(6)
(2)
(c)
(3)
(d)
First Reduction Equation
+516959
- 25907
+ 130/,
+ 35753
-102131
+ 5130
+ 15707
+ SO 177
+ 23492
+ 59125
- 2.970
- 38804
- 11681
- 22915
4- 6763
+ 40826
472048
23711
32272
93258
96355
53988
6751 1
(4)
-.05023
.19756 +.114.37 - .91313
The above equations are treated in a similar manner as the
original normal equations, and we have therefore the 2d reduc-
tion equation of the form:
(a)
(1)
(&)
(2)
Second Reduction Equation
+ 34451
+ 10577
- 35834
+
8561
+ 32 /t7
- 11002
+
2628
+ 3315
- 11234
+
3097
+ 37273
—
8905
+ 40064
—
13523
+ .30702
-1.04014
+
.24850
: Equation
+ 08
232
+
469
+ 791
-
1600
+ 2791
—
4618
(3)
(a)
(1)
(2)
-3.41170 +6.8.9706
Fourth Reduction Equation
+ 2000 - 3018
The solution for the unknown r's may now be shown as fol-
lows:
re =3018:2000 = 1.5090
ro= -6.8971 -(1.5090)( -3.4117) = -1.7488
n=- 0.2485 - ( - 1.7488) (0.3070) - (1.5090) ( - 1.0401) = 1.8580
ra = 0.9131- (1.8580)(-0.0502)-(-1.7488)(-0.1976)- (1.5090)
(0.1144) =0.4884
ro -- 1.1666 - (0.4884) ( - 0.0002) - (1.8580) (0.0805) - ( - 1.7488)
( -0.0047) - (1.5090) ( -0.0394) = 1.0679
232 DETERMINATION OF PARAMETERS [121
From the above values of r and by means of the relation
A•,= ,^A^2for^=0, 3,4, 5and6
we can easily determine the most probable values of A;, with
which the original observation equations as shown on page (228)
must be multiplied so as to satisfy the observed values of F{x)
in the sense as implied in the method of least squares.
This results in the following arrangement:
z
A-o<^o(^)
A-3^3(Z)
k-i<Pi{2)
A-5^5(-')
A-6^6(2)
^ki<Pi
Obs.
-0.688
74.6
+26.7
+ 4.4
+ 19.7
+ 7.2
132.6
133
+0.402
87.2
-20.5
+28.1
-17.2
— 22.4
55.2
55
1.493
31.0
- 7.3
-26.2
+ 3.9
+21.5
22.9
23
2.583
3.4
+ 6.6
+ 3.9
+ 2.3
- 9.1
7.1
7
3.674 0.2 + 0.9 +1.8 - 0.9 + 1.0 3.0 2
+4.764 0.0 + 0.0 + 0.1 + 0.0 + 0.0 0.1 2
The agreement between the calculated values and the origi-
nally observed series leaves evidently little to be desired in the
way of a satisfactory "fit."
If we limited ourselves to three terms of the series and put
F(x) = (p{z)^ ^knpiiz) for « = 0, 3 and 4
and then determined kn, ky, and A;4 b y the method of least squares
the final result would be of the f oi'm :
<r(2) =264.2(^0(2) -89.9sr;,(z) -5.2(^^(2),
for which the calculated values of the frequency function would
be as follows:
X
F(x)
Obs.
Pearson
5
131.6
133
136.9
6
55.2
55
48.5
7
24.5
23.
22.6
8
15.5
7
9.6
9
1.6
2
3.4
10
0.2
2
0.8
The fit is evidently not so close as when we use 6 terms, but
it is by no means a poor fit and does not require nearly so much
arithmetical work as the larger number of terms in the fre-
quency series. In this connection it is of interest to compare
the present graduation with the result reached by Pearson,
which is also shown in the above table. Taken all in all there
121] DETERMINATION OF PARAMETERS 233
is no doubt in my mind that the serial expension gives far better
results than the Pearsonian methods and does not entail nearly
so much labour as these.
Note on Adjusted Moments. — The theoretical moments are given
in the form of definite integrals while the observations always give us the
moments on the form
n= ■» n + l'ia
n= X n — ^-^a
where a is the class interval of the observations. In order to determine
the semi-invariants it is, however, required to know the continuous
moments
Mr =fx'<P{x)dx
The values of s being given in the form of finite sums and not as defi-
nite integrals are therefore subject to certain adjustments if we wish to
express them as continuous moments. The necessary adjustments can,
however, easily be performed by well-known formulas from the theory
of mechanical quadrature if the frequency function and its derivatives
vanish for x = — =» and x = + » . The English mathematician, Sheppard,
has among others developed the following simple formulas for the transi-
tion from s to M :
il/o =So, Ml =Sl, Mi =S2 — T^S2. Mi =S3 —7 Si
,r a2 7« 4 r)«2 7a2
A/4 =S4 — -J so I .7T7) *'"' '"5 ~^'^ — (^ *'3 +4© *i
The Sheppard adjustments again emphasizes the fact that the method
of moments works with curve areas instead of curve ordinatcs, which
necessarily must lead to some sort of mechanical quadrature formula
unless we are able to evaluate the indefinite integrals of the expressions
for the frequency functions. If we use curve ordinates to calculate the
specific numerical values of Pearson's frequency functions we are liable
to encounter large errors. This fact is among other things pointed out
by Caradog Jones who in mentioning the use of ordinates points out that
"it must be remembered the resulting values are only a first approxima-
tion to the observed frequencies and a better series is obtained if, by
using some good quadrature formula, we calculate the AREAS for the
suecessi\'e groups between the curve, the l)ounding ordinates, and the
axis of X." This is one of the great drawbacks to the otherwise elegant
Pearsonian types of frequeiKn' curves because it entails a large amount
of arithmetical work to compute specific numerical values from the final
formulas as determined by Pearson's curve types.
Any reader who will take the trouble to consult the original memoirs
by Pearson and Elderton and the recently published treatise by Mr.
234 DETERMINATION OF PARAMETERS [121
Caradog Jones will tlirro find ample evidence of the large amount of
tedious arithmetical work involved in the application of mechanical
quadrature formulas. The recently suggested finite difference equation
formulas by the American mathematican, Carver, while emphasizing the
ditliculty of applying the Pearson system, do not tend materially to shorten
the arithmetical work very much, and Mr. Carver must in the final in-
stance resort to mechanical quadratiu*e. *
All these difficulties are, however, eliminated in the case of the deter-
mination of the frequency function in serial form by means of the method
of least squares where we work equally well with ordinates as with areas.
A firrther advantage of the Gram — Charlier expansion in series is found
in the fact that standard tables of the generating function and its deriva-
tives as well as the definite integrals of these functions have been published
both by Briihns, Charlier and Jorgensen. Speaking from a purely per-
sonal point of view I wish to state that tlu-ough a long and varied experi-
ence in i>ractical curve fitting to the most diverse kinds of statistical data
I have had occasion to use both the Pearsonian and the Gram — Charlier
tyi)e of curves, and while I fully recognize the theoretical elegance and
apparent simplicity of the Pearson system, I feel nevertheless that from
the point of view of the practical computer the older system as devised
by the Scandinavian investigators is to be preferred in comparison with
the methods advocated by the followers of the distinguished founder of
the English Biometric school.
*Mr. Carver s able anri intorfslinp; analysis by means of finite diderence equations
is, however, to a Kreat extent antcccded by Iho much earlier Danisii memoirs of Opper-
man and dram where the finite diirin^nee e(|uation methods arc diseussed.
CHAPTER XVI
LOGARITHMICALLY TRANSFORMED FREQUENCY
FUNXTIONS
122. Transformation of the Variate. — While it is always
possible to express all frequency curves by an expansion in
Hermite polynomials, the numerical labor when carried on by
the method of least squares often involves a large amount of
arithmetical work if we wish to retain more than four or five
terms of the series. Other methods lessening the arithmetical
work and making the actual calculations comparatively simple
have been offered by several authors and notably by Thiele,
who in his works discusses several such methods. Among
those we may mention the method of the so-called free func-
tions and orthogonal substitution, the methods of correlates
and the adjustment by elements. The chapters on these
methods in Thiele's work are among some of the most import-
ant, but also some of the most difficult in the whole theory of
observations and have not always been understood and appre-
ciated by the mathematicians, chiefly on account of Thiele's
peculiar style of writing. A close study of the Danish scholar's
investigations is, however, well worth while, and Thiele's work
along these lines may still in the future become as epochmaking
in the theory of probability as some of the researches of the
great Laplace. The theory of infinite determinants as used by
M. Fredholm in the solution of integral equations is another
powerful tool which offers great advantages in the way of
rapid calculation. All these methods require, however, that
the student must be thoroughly familiar with the difficult
theory upon which such methods rest, and they have for this
reason been omitted in an elementary work such as the present
treatise.
We wish, however, to mention another method which in the
majority of cases will make it possible to employ the Gram or
Laplacean — Charlier curves in cases with extreme skewness
or excess. We have here reference to the method of logarithmic
transformation of the variate, x.
235
236 TRANSFORMED FREQUENCY CURVES [124
123. The General Theory of Transformation. — ^One of
the simplest transformations is the previously mentioned linear
transformation of the form z^f{x) =ax -(- h, by which we can make
two constants, Ci and ci vanish. Other transformations sug-
gest themselves, however, such as f{x) =ax--\-bx-\-c, f{x) = V x,
f{x) =logx and so forth. For this reason I propose to give a
brief developm.ent of the general method of transformations of
the statistical variates, mainly following the methods of Char-
lier and J0rgensen.
Stated in its most general form our problem is: If a fre-
quency curve of a certain variate is given by F{x) what will be
the frequency curve of a certain function of x, say /(a:)?
The equation of the frequency curve is y^F{x), which
means that F{x)dx is the probability that x falls in the interval
between x — Yzdix and x-\-'^-?d^- The probability that a new
variate z after the transformation z =f{x), ovx(z) =x, falls in the
interval z — }4dz and z-\- y^dz is therefore simply
F[x{z)]x\z)dz = F{x)dx,
which gives in symbolic form the equation of the transformed
frequency curve.
The frequency for z =f(x) is of course the same as for x. The
ordinates of the frequency curve, or rather the areas between
corresponding ordinates, are therefore not changed, but the
abcissa axis is replaced by f{x). Equidistant intervals of x
will therefore not as a rule — except in the linear transformation
— correspond to equidistant intervals of /(z).
If, for instance, the frequency curve F{x) is the Laplacean
normal curve
1 _ j-2 --zcri
F(x)= — ^e ■
o-V27r
and if we let z =f(x) =x- or x = \ z, we have evidently
1 e-^:-2<^^
F(z)=— -—
ayj 2t 2\ Z
124. Logarithmic Transformation. — Of the various trans-
formations the logarit h mlc is of special importance. 1 1 happens
that even if the variate ./• forms an extremely skew frequency
distribution its logarithms will be nearly normally distributed.
124] LOGARITHMIC TRANSFORMATION 237
This fact was already noted by the eminent German psycholo-
gist, Fechner, and also mentioned by Bruhns in his Kollek-
tivmasslehre. But neither Fechner nor Bruhns have given a
satisfactory theoretical explanation of the transformation and
have limited themselves to use it as a practical rule of thumb.
Thiele discusses the method under his adjustment by ele-
ments, but in a rather brief manner. The first satisfactory
theory of logarithmic transformation seems to have been given
first by J0rgensen and later on by Wicksell.* J0rgensen first
begins with the transformation of the normal Laplacean fre-
quency curve. Letting z = logx and bearing in mind that the
frequency of x equals that of logx we have
z =f(x) = logx, or x = x {z) = e' ami dx — e'dz
The continuous power sums or moments of the rth order
around an arbitrary origin take on the form
/-(- >i 1 / J — m\.2 f^x Ifl'x.li — m\'l
x'-e -^ " ' dx={n\' 27r)-wJ x'e-^~''~''^ dx
— X o
r + ^ -:^('—"lY
The change in the lower limit in the second integral from
— 3c to zero arises simply from the fact that the logarithm of
zero equals minus infinity and the point — » is thus by the
transformation moved up to zero.
By a straightforward transformation (see appendix) we may
write the above integral as
M,= e e-'''"dt = Ne
f^
\ 27r
Changing from moments to semi-variants by means of the
well-known relations
AO — Mo
\i = Mi:Mo
l2 - (M 2M0 - M,-) :Mo'
*Thelaw of errors. leadiiiK to the geometric mean as the most probable value of the
vanate as discovered by Sir Donald McAHster in 1879 mav, however, be considered
as a toreruiu.er ot Jiirgeiisen's work.
238 TRANSFORMED FREQUENCY CURVES [125
X3 = (M3Mo=- 3M2MiMo+2Mi^):Mo'
we have
Ai — e
X, = e-"'+-"'\e"'-l)
These equations give the semi-invariants expressed in terms
of m and n. On the other hand if we know the semi-invariants
from statistical data or are able to determine these semi-in-
variants by a priori reasoning we may find the parameters m
and n.
125. The Mathematical Zero. — A point which we must
bear in mind is that the above semi-invariants on account of
of the transformation are calculated around a zero point which
corresponds to a fixed lower limit of the observations.
Very often the observations themselves indicate such a lower
limit beyond which the frequencies of the variate vanish
In the case of persons engaged in factory wj.'k t'i3.'3 is in most
countries a well-defined legal age limit below which it is illegal
to employ persons for work. Another example is of erad in
the number of alpha particles radiated from certain radioa3tive
metals. Since the number of particles radiated in a certain
interval of time must either be zero or a whole positive number
it is evident that —1 must be the lower limit because we can have
no negative radiations. Analogous limits exist in the age
limit for divorces and in the amount of moneys assessed in the
way of income tax.
The lower limit allows, however, of a more exact mathe-
matical determination by means of the following simple con-
siderations. It is evident that this lower limit must fall below
126] MATHEMATICAL ZERO 239
the mean value of the frequency curve. Let us suppose that
it is located at a certain point, a, at a distance of yj units from
the mean M = /.i{x) — y; = a; and let us furthermore as a be-
ginning place the origin at Xi(x), in which case Xi of course
equals zero. By shifting the origin to a, which implies a
translation of >7 units in negative direction, the original variate
(x) is transformed into x + y;^ and Xi will now equal rj while
the semi-invariants of higher order remain the same as before
the transformation because of the well known relation
Xr(a: — >:) = XrC^) for r >1
We may therefore write the previously given relations
between the X's and m and n as follows:
X2 = yi-(e"'-l) or e"' = l+Xo:-,72
which reduces to /^syj^ — S/^2-yj- — /i'^^ = 0.
The solution of this cubic equation which has one real and
two imaginary roots gives us the value of >: or Xi — a and thus
determines the mathematical zero or lower limit. We have in
fact:
n^ = log( 1 + X-i :y;~) and
w = log -<■ — 1.5«-, while
126. Logarithmically Transformed Frequency Series. —
We have already shown that the generalized frequency curve
could be written as
F{x)=c,<f,{x)- -^j +~— gj-+
where the Laplacean probability function
■ ^o{x)=^==e ^"^
is the generating function with M and a as its parameters.
240 TRANSFORMED FREQUENCY CURVES [126
The suggestion now immediately arises to use an analogous
series in the case of the logarithmic transformation. In this
case the frequency curve, F{x), with a lower limit would be ex-
pressed as follows :
„, , , , . ki'i'iix} , k-i'l-iix) ks^six) .
Fix) =koi'o{x) 1 -,
1! 2! 3!
while the generating function now is
nV 2ir
where m and 7i are the parameters.
Using the usual definition of semi-invariants we then have
Xico + Xoj 2 -f-XgajS-f-
TT 2r ^r SiOJ S2C0- Ssco''
o
The general term on the right hand side integral is of the form
/»
o
where the integral may be evaluted by partial integration as
follows:
e""* ,(a;) dx = e'^'^.-.ix) ] - co J e""*,-i(x) dx
o ^ o •
Since both <I>o(x) and all its derivatives are supposed to
vanish for x=0 and a: = oo the first term to the right becomes
zero and
By successive integrations we then obtain the following
recursion formula
126] LOGARITHMIC TRANSFORMATION 241
o o
o
O O
Or finally
o
Expanding e^'^ in a power series we have
■V ^TT
o o
The general term in this expansion is of the form
V^2- r! J '
o
' x'^e "'■ " -• dx
which according to the formulas given on page (237) reduces to:
(_^)-'e'"''"+'^+'^"'^'-+'^V:r!
Hence we may write
r = 00
/ e^'^^J,x)dx = {-o)y y'jB ' ^ '^'- ' ^ ' co'':r!
o r =0
Consequently the relation between the semi-invariants and
the frequency function
Fix) =kofo{x) -^\4^x(x) +^'*2(x) --%3(x) + . . . .
242 TRANSFORMED FREQUENCY CURVES [126
can be expressed by the following recursion formula
-If —2!" "31 , Sioj 820;- S?a)
V =0 s = o r = o
The constants k are here expressed in terms of the unadjusted
moments or power sums, s. It is readily seen that the Sheppard
corrections for adjusted moments, M, also apply in this case.
We are, therefore, able to write down the values of the k's from
the above recursion formula in the following manner
M4 = A:4e'"+^'^"' +4^:36-'"+-"' +6A:2e^'"+*-^"' +4ii:ie'*'"+^'^'
It is easy to see that it is not possible to determine the gener-
ating function's parameters m and n from the observations.
These parameters like M and rr in the case of .the Laplacean
normal probability curve must be chosen arbitrarily. If
m and n are selected so as to make A;i and kz vanish we have
M2 = A;oe=''"+*-^"'
the solution of which gives
^n2 ^ M.M, ,^ _ _M7_ ^ Mo'M2
while
127] PARAMETERS DETERMINED BY LEAST SQUARES 243
This theory requires the computation of a set of tables of the
generating function
1 irlog x—mT^
and its derivatives. For ^\\){x) itself we may of course use the
ordinary tables for the normal curve <i?o(2:) when we consider
log x — m
z =
n
I have calculated a set of tables of the derivatives of <to(x)
and hope to be able to publish the manuscript thereof in the
second volume of this treatise.
127. Parameters Determined by Least Squares. — The
above development is based upon the theory of functions and
the theory of definite integrals. We shall now see how the
same problem may be attacked by the method of least squares
after we have determined by the usual method of moments the
values of m and n in the generating function <^o(2:).
Viewed from this point of vantage our problem may be
stated as follows:
Given an arbitrary frequency distribution, of the variate z
with z = ([og X — m) m and where x is reckoned from a zero
point or origin a, which is situated r, units below the mean and
defined by the relation
r]^\:i — Zy]'-\'f = \2, where -n^Xv-a;
to develop F{z) into a frequency series of the form
F{z) =k^<Pi){z)+k:i^:i{z)-\-kA<Pi{z)+ . . , .-\-kn<Pn(z),
where the k's must be determined in such a way that the ex-
pression
i = n
gives the best approximation to F{z) in the sense of the method
of least squares.
Stated in this form the frequency function is reduced to the
ordinary series of Gram or the A type of the Charlier series,
already treated in the earlier chapters.
244 TRANSFORMED FREQUENCY CURVES [128
128. Application to Graduation of a Mortality Table. —
As an illustration of the theory to a practical problem we pre-
sent the following frequency distribution by 5-year age intervals
of the number of deaths (or -c?,. by quinquennial grouping)
in the recently published American-Canadian Mortality of
Healthy Males, based on a radix of 100,000 entrants at age 15.
Frequency Distribution of Deaths by Attained Ages in
American-Canadian Mortality Table
Ages
^dx
1st Component
2d Component
15- 19
1,801
120
1,681
20- 24
1,996
230
1,766
25- 29
2,089
440
1,649
30- 34
2,120
790
1,330
35- 39
2,341
1.370
971
40- 44
2.911
2,270
641
45- 49
3,937
3,570
367
50- 54
5,527
5,400
127
55- 59
7,723
7,722
1
60- 64
10,383
10,383
65- 69
12,987
12,987
70- 74
14.535
14,535
75- 79
13,807
13,807
80- 84
10,328
10,328
85- 89
5,464
5,464
90- 94
1,757
1,757
95- 99
278
278
00-104
16
16
100,000 91,467 8,533
The curve represented by the c/.^ column is evidently a com-
posite frequency function compounded of several series. From
a purely mathematical point of view the compound curve may
be considered as being generated in an infinite number of ways
as the summation of separate component frequency curves.
From the point of view of a practical graduation it is, however,
easy to break this compound death curve up into two separate
components. A mere glance at the d^ curve itself suggests a
major skew frequency curve with a maximum point somewhere
in the age interval from 70 75 and minor curve (practically
one-sidedj for the younger ages.
Let us therefore break the -dj. column up into the two so far
perfectly arbitrary parts as shown in the above table and then
try to fit those two distributions to logarithmically transformed
A curves.
128] GRADUATION OF A MORTALITY TABLE 245
Starting with the first component the straightforward com-
putation of the semi-invariants is given in the table below with
the provisional mean chosen at age 67.
Frequexcy Distribution of Deaths in Ameriean Mortality Table
First Component
Ages X F(x) xF(x) xiF(x) x^F(x)
104-100
— 7
16
112
784
5,488
99- 95
- 6
278
1,668
10,008
60,048
94- 90
— 5
1,757
8,785
43,925
219,625
89- 85
- 4
5,464
21.856
87,424
349,696
84- 80
- 3
10,328
30,984
92,952
278,856
79- 75
- 2
13.807
27,614
55,228
110,456
74- 70
- 1
14,535
14,535
14,535
14,535
69- 65
-
12,987
V
59,172
105,554
304,856
1,038,704
64- 60
+ 1
10.383
10,383
10.383
10,383
59- 55
+ 2
7.723
15,446
30,892
61,784
54- 50
+ 3
5,400
16.200
48,600
145,800
49- 45
+ 4
3.570
14,280
57,120
228,480
44- 40
+ 5
2,270
11,350
56,750
283,750
39- 35
+ 6
1,370
8.220
49,320
295,920
34- 30
+ 7
790
5.530
38,710
270,970
29- 25
+ 8
440
3.520
28,160
225,280
24- 20
+ 9
230
2.070
18,630
167.670
19- 15
+ 10
120
1.200
12,000
120.000
V
^
32,296
88.199
350.565
1,810,037
sr 91,468 -17.355 655,421 771,333
Computing the semi-invariants by means of the usual for-
mulas in paragraph 104, we have:
/.,= -17355:91468= -0.18974,or mean at age 67+5(0.19) or at
age 67.95
/.2 = 655421 :91468 - /-i'- = 7.1296
Xa = 771333:91468 -3>-,>-+2X,'' = 12.4981
In order to determine the mathematical zero or the origin
we have to solve the following cubic:
/'^i'/:'* — o'/.-rc = X2'\ or
12.489-/:' - 152.511-/:- = 362.47
the positive root of which is equal to 12.39. The zero point
is therefore found to be situated 12.39 5-year units from the
mean or at age 67.95 + 5(12.39), i. e. very nearly at age 130,
246 TRANSFORMED FREQUENCY CURVES [129
which we henceforth shall select as the origin of the co-ordinate
system of the first component. We have furthermore
12.39 = e"'^^-^"', and 1.1296 = e~"''--"'\e''' -1) =(12.39)-^(e"'-l),
the solution of which gives rz- = 0.04436, ?i = 0.2106, m = 2.4504,
all on the basis of a 5-year interval as unit. If we wish to
change to a single calendar year unit we must add the natural
logarithm of 5, or 1.6094, to the above value of m, which gives us
7;; =4.0598, while n remains the same. The above computa-
tions furnish us with the necessary material for the logarithmic
transformation of the variate x which now may be written as
2 = [log (130 -x)- 4.0598] :0.2106,
where x is the original variate or the age at death.
Having thus accomplished the logarithmic transformation
we may henceforth write the generating function as
1 lr log(130-x)-4.05 9812 -i
^ (^\ — ^L 0.2106 J /^\ -.-^^-^
.2106V 27r ^ V27r
We express now F{x) by the following equation.
Fix) =ko'io{x) -\-k:i^:i(x) -\-h^i{x) -{- ....
or in terms of the transformed z:
<piz) =h)(P){z) -{-hip-iiz) ^ki<Pi(z) -It . . . . ,
and proceed to determine the numerical values of k by the
method of least squares.
129. Formation of Observation Equations. — The values
of <po{z) and its 3rd and 4th derivatives may be written down
directly from the tables of J0rgensen or Charlier for various
values of z as shown in detail in the following scheme on
the following pages.*
<l)
(-')
{'■'.)
(4)
(5)
(6)
(7)
(8)
(0)
Agf
z
(fltlZI
ip:i(z)
<P4,(Z)
koVi)
A-;i(4)
^•4(5)
Fiiz)
1.5
+:i.257
+ .0020
-.0491
+ .1029
+ 14
+ 10
- 1
23
:^.2\:^
.002:',
.0.537
.10S4
17
10
1
20
7
:',.\7()
.002(;
.O.ISG
.1140
19
12
I
30
8
3.127
.oo:]o
.0()37
.1208
22
14
1
3.-)
*The values of z, co-ordinate wit li lliose of x arc computed for ail integral values of
X from 1.5 to 100 in a(-cordanre with the previously established relation, viz. ^=|log
(130-z)-4.0(i»8l:0.2106. All logarithms on base e.
129]
OBSERVATION EQUATIONS
247
(1)
(2)
(3)
(4)
(5)
(0)
(7)
(8)
(9)
Agex
z
^o(^)
(^3(2)
^^4(2)
Ao(3)
A;3(4)
/b4(5)
Fi(z)
9
3.085
.0034
.0688
.1249
25
15
39
20
3.041
.0039
.0744
.1290
29
16
44
1
2.999
.0044
.0803
.1331
32
17
48
2
2.955
.0051
.0850
.1361
38
18
57
3
2.911
.0057
.0919
.1382
42
19
60
4
2.866
.0065
.0981
.1391
48
21
68
25
2.821
.0074
.1044
.1390
54
22
75
6
2.776
.0085
.1104
.1367
63
24
86
7
2.730
.0096
.1168
.1328
71
25
86
8
2.683
.0110
.1229
.1264
81
26
10()
9
2.637
.0123
.1286
.11.39
91
27
117
30
2..587
.0140
.1340
.1072
103
28
130
1
2..542
.0150
.1387
.0943
116
29
144
2
2.494
.0178
.1420
.0763
131
30
160
3
2.445
.0201
.1462
.0.376
149
31
179
4
2.396
.0226
.1486
.0340
166
32
19S
35
2.346
.0255
.1496
+ .0039
188
32
-
220
6
2.296
.0286
.1489
-.0275
210
32
+
242
7
. 2.245
.0320
.1464
.0622
236
31
1
268
8
2.193
.0360
.1423
.0983
265
30
1
296
9
2.142
.0402
.1393
.1399
296
29
1
326
40
2.089
.0450
.1281
.1864
331
27
2
260
1
2.036
.0502
.1170
.2355
369
25
2
39()
2
1.982
.0559
.1030
.2875
411
22
3
43(>
3
1.928
.0621
.0859
.3412
452
18
3
47S
4
1.873
.0690
.0656
.3965
508
14
4
526
45
1.822
.0757
.0442
.4474
557
9
4
570
6
1.762
.0845
-.0156
..3060
622
+ 3
5
630
7
1.704
.0934
+ .01.34
..3596
687
- 3
6
690
8
1.W7
.1028
.0487
.6082
7.38
10
6
7.54
9
1.589
.1129
.0853
.6419
832
18
6
820
50
1.529
.1239
.12.55
.()893
913
27
7
893
1
1.471
.1352
.1599
.7132
994
34
967
2
1.409
.1479
.2114
.7349
1,089
45
1.051
3
1.348
.1609
.2565
.7430
1,185
54
1,138
4
1.286
.1745
.3022
.7307
1,288
63
1,231
55
1.224
.1886
.3467
.7062
1,391
74
1,324
6
1.160
.2035
.3907
.6642
1.501
83
1,425
7
1.095
.2190
.4320
.6037
1,612
92
6
1,526
8
1.030
.2347
.4688
.5257
1,730
99
5
1,636
9
0.963
.2509
.5008
.4180
1,847
106
4
1,745
60
0.896
.2672
.5257
.2911
1,965
112
3
1,856
1
0.828
.2832
.5426
.1831
2,083
115
2
1,970
2
0.758
.2994
.5489
-.03.30
2,201
lit)
+
2,085
3
0.689
.3146
..3474
+ .1187
2,318
116
- 1
2,201
4
0.617
.3298
.5329
.2839
2,428
113
3
2,312
248
TRANSFORMED FREQUENCY CURVES
[129
(1)
(2)
(3)
(4)
(5)
Agtv
r
(^oU)
(fai-)
^4(Z)
65
0.543
.3443
.5056
.4537
()
0.470
.3572
.4666
.6156
7
0.396
.3689
.4152
.7686
8
0.319
.3792
.3505
.9098
9
0.243
.3873
.2768
1 .02()2
70
0.164
.3937
.1918
1.117t)
1
.084
.3975
.0999
1.17.")7
2
+0.000
.3989
+ .0119
1.1968
3
-0.080
.3976
-.0952
1.1777
4
0.164
.3937
.1918
1.1176
75
0.249
.3868
.2829
1.0180
()
0.348
.3755
.3762
.8592
7
0.425
.3645
.4368
.7043
8
0.516
.3493
.4912
.5146
9
0.608
.3316
.5303
.3069
80
0.702
.3118
.5502
+ .0892
1
0.798
.2902
.5473
-.1204
2
0.896
.2672
.5257
.3130
3
0.996
.2436
.48,59
.4380
4
1.098
.2185
.4302
.5899
85
1.203
.1934
.3614
.6943
6
1.309
.1694
.2854
.7358
7
1.418
.1460
.2048
.7340
8
1.529
.1240
.1255
.(5893
<)
1.644
.1034
-.0505
.6106
!K)
1.7(>2
.0845
+ .01.56
.5060
I
1 .882
.0679
.0693
.3874
2
2.004
.0536
.1090
.2663
3
2.132
.0397
.1380
.1485
4
2.260
.0310
.1478
-.0483
95
2.393
.0227
.1477
+ .0325
()
2..-)30
.0163
.1399
.0905
7
2.()73
.0100
.1207
.1295
8
2.S21
.0074
.1044
.1386
9
2.9»).S
.0050
.0842
.1353
100
-3.124
.0028
.0640
.1203
(6) (7) (8) (9)
A-o(3) k3{4) M5) Fi(z)
2,532 107 5 2,420
2,627 99 6 2,522
2,716 88 8 2,620
2,789 74 9 2,706
2.848 59 10 2,779
2,900 41 11 2,848
2,929 21 12 2,896
2,937 - 3 12 2,922
2,929 + 20 12 2,937
2,900 41 11 2,930
2,848 60 10 2,898
2,767 80 9 2,848
2,686 93 7 2,772
2,569 104 5 2,668
2,444 112 3 2,553
2,296 117 - 1 2,412
2,134 116 + 1 2,251
1,965 112 3 2,080
1,788 103 4 1,895
1,612 91 9 1J09
1,420 77 7 1,504
1,244 60 7 1,311
1,075 43 7 1,122
913 27 7 947
758 +11 6 775
622 - 3 5 624
500 15 4 489
394 23 3 374
292 29 1 264
228 31+0 197
167 31 - 135
120 30 1 89
76 26 1 49
54 22 1 29
37 18 1 18
21 14 1 6
Since the original observations of d^ are given in 5-year age
intervals it becomes necessary to sum the numerical values of
ipi,{z) and its derivatives by quinquennial age groupings so as to
form the required observation equations. We find thus for
instance in the age interval 55-59 the following observation
equation (the summation to take place from x = 55 to x = 59)
hl<pi,(z) +hlip:i{z) +ki^(p4(z) =^ip{z) =0^, or
1.0967A:o-h2.1390fc3 -2.9178^4 = 7722
129]
OBSERVATION EQUATIONS
249
Similar equations are formed for the other age intervals, re-
sulting in the following tabular representation of the coefficients
to the various k's and the observed values of the frequency dis-
tribution, (p(z)
TABLE I
Tabular Arrangements of Numerical Data in the Observation
Equations
Ages
(po
«P3
<P4
15-19
.0130
- .2930
+ .5730
120
20-24
.0256
- .4305
+ .5758
230
25-29
.0488
- .5833
+ .6508
440
30-34
.0903
- .7104
+ .3694
790
35-39
.1623
- .7265
- .3240
1,370
40-44
.2822
- .4996
-1.4471
2,270
45-49
.4673
+ .0896
-2.7631
3,570
50-54
.7424
+ 1.0555
-3.6111
5,400
55-59
1.0967
+2.1390
-2.9178
7,722
60-64
1.4942
+2.6975
- .1066
10,383
65-69
1.8369
+2.0147
+3.7739
12,987
70-74
1.9814
+ .0166
+5.7854
14,.535
75-79
1.8077
-2.1174
+3.6030
13,807
80-84
1.3307
-2.5393
-1.3721
10,328
85-89
.7362
-1.0276
-3.4(>40
5,464
90-94
.2729
.4.571
-1.3925
1,757
95-99
.0609
.5714
.6125
278
100-
.0068
.1714
.3890
16
From the above table we notice that we have 18 observation
equations from which to determine the three unknown para-
meters ko, h and ki. The number of equations being greater
than the number of unknowns we make use of the method of
least squares. While a direct application of this principle of
course is feasible, it will, however, be found easier to start with
an approximate solution for kn, ki and A-) and then apply the
method of least squares. It will be found that in the three age
intervals 6L-69, 70-74 and 75-79 where the observations are most
numerous the observations will be approximately satisfied by
the following preliminary values of k, viz. :
A:io = 7300, A;'3= -340 and kU= -50.
Multiplying the above values of k with their respective
columns in Table I, or in other words forming the products
A:Vo, kh<p A and k^ft, we obtain a new table of the following form :*
*l.asl figure omitted.
250 TRANSFORMED FREQUENCY CURVES [129
TABLE
II
Control
Column
Ages
a
b
c
s
15- 19
10
10
- 3
- 12
5
20- 24
19
15
- 3
- 23
8
25- 29
36
20
- 3
- 44
9
30- 34
66
24
- 2
- 79
9
35- 39
119
25
2
- 137
9
40- 44
206
17
7
- 227
3
45- 49
343
- 3
14
- 357
- 3
50- 54
542
-36
18
- 540
-16
55- 59
801
-73
15
- 772
-29
60- 64
1,091
-92
1
-1038
-38
65- 69
1,341
-69
-19
-1299
-46
70- 74
1,446
- 1
-29
- 1454
-38
75- 70
1,320
72
-18
-1381
- 7
80- 84
971
86
7
-1033
31
85- 90
537
35
17
- 546
43
90- 94
202
-16
7
- 176
17
95- 99
45
-20
- 3
- 28
- 6
100-104
5
- 6
- 2
2
- 5
9,100 -12 6 -9148 -54
The formation of the sum-products [aa], [ah], .... [ao],
[66], [6c], [bo] and [cc], [co] proceeds now in routine manner as
shown in the following tables:
TABLE III
[aa] [ah] [ac] [ao] [as]
100 100 - 30 - 120 50
3()1 285 - 57 - 437 152
1,296 720 - lOS - 1,584 324
4,350 1,584 - 132 - 5,214 594
14,161 2,975 23S - 16,303 1,071
42,436 3,.-)02 1,442 - 46,762 618
117,649 - 1,029 4,802 - 122,451 - 1,029
293,764 - 19,512 9,756 - 292,680 - 8,672
641.601 - 5S,473 12,015 - 618,372 - 23,229
1,190,281 -100,372 1,091 -1,132.458 - 41,458
1,798,2S1 - <»2,.-)29 -25,479 - 1,741. 9.")9 - 61,686
2,090.916 - 1.116 -41,934 -2,1()2,4S4 - 54,948
1,742.400 95,010 -23,760 -l.,S22,92() - 9,240
942,841 83,5()(i (j,797 -1,003.043 30,101
288,369 18,795 9,129 - 293,202 23,091
40,804 - 3,232 1,414 - 35,552 3,434
2,025 - 900 - 135 - 1,260 - 270
25 - 30 10 - 10 - 25
9,211,666 - 71,016 -44,961 -9,236,811 -141,122
129] OBSERVATION EQUATIONS 251
TABLE IV
[66j
lbc\
\bo]
[b,\
100
- 30
- 120
50
225
- 45
- 345
120
400
- 60
- 880
ISO
576
- 48
- 1,896
216
625
50
- 3,425
225
289
119
- 3,859
51
9
- 42
1,071
9
1,296
- 648
19,440
576
5,329
-1,095
56,356
2,117
8,464
- 92
95,496
3,496
4,761
1,311
89,631
3,174
1
29
1.454
38
5,184
-1,296
-99,432
- 504
7,396
602
-88,838
2,66()
1,225
595
-19.110
1,505
256
- 112
2.S16
- 272
400
60
560
120
36
12
12
;;o
36,572
[cc]
9
9
9
4
4
49
196
324
225
1
361
841
324
49
289
49
9
4
- 690
TABLE V
\co]
36
69
132
158
- 274
- 1.5S9
- 4,998
- 9,720
-11,580
- 1,038
24,681
42,166
24,858
- 7,231
- 9,282
- 1,232
14
4
48,931
["•1
- 15
- 24
- 27
- 18
IS
21
- 42
- 288
- 435
- 38
874
1,102
126
217
731
119
18
10
13,797
2,756 45,244 2.34.)
From the above tables we may now write down the following
scheme for the solution of the normal equations by means of
the Gaussian algorithmus
252
TRANSFORMED FREQUENCY CURVES
[129
TABLE VI
Scheme for tiik Solution of Normal Equations
921,107 - 7102 - 4496 -923681
55
3657
35
69
22
276
7122
4893
4508
4525
-.00771
3602
-.00488
- 104
3
254
-1.00273
- 2229
64
16
-.02887
- .61882
251
48
Solving for the unknowns we have now
r4= 48:251 -.19123
rs = .61882 - (.19123) ( - .02887) = .62434
ro = 1.00273 - (.62434) ( - .00771) - (.19123) ( - .00488)
= 1.00847
We therefore find the following numerical values for the
probable values of ku, k.i and k\
A-o-U-'o = (1.00847) (7300) = 7,361.8
k, = r,k', = (.62434)1-340) = -212.2
A;., = r.,A;'4 = (.19123)(- 50) = - 9.6
The next step is then to form the three columns A:ii(yro( 2),
k3(p:iiz) and k^ip^iz) for the individual ages from 15 and upwards.
The formation of -k,if,(z) gives us finally the separate values
by integral ages of the first component curve or Fi{.r).
If we now subtract this component from the originally ob-
served values of the compound curve, d,., we obtain the follow-
ing values (arranged in quinquennial age groups) for the second
component, Fii{x).
129] OBSERVATION EQUDTIONS 25i
Ages /''ii(x)
0- 4 ;■■)() )
5- 9 440 V Hvpothotical Values
10 14 1,210 J
15-19 l,()r)0
20-24 1,719
25-29 l.()10
30-34 1,309
35-39 989
40-44 715
45-59 473
50-54 247
55-59 67
10,479
It is of course possible to fit this particular curve type directly
to a logarithmically transformed Gram or Charlier A type of
frequency curves, although this will require 5 or 6 terms of the
series. But still greater obstacles would be encountered if we
were to attempt a graduation by means of Pearsonian curve
types. The goal can, however, be reached quite readily by the
introduction of a certain hypothetical device. Any reader
familiar with the various types of frequency curves will readily
notice that the above frequency distribution of Fuix) repre-
sents a truncated curve from which the curve segment corres-
ponding to ages below 15 has been eliminated. We may there-
fore substitute the following, so far hypothetical values for the
missing curve segment. For ages 0-4 the value of 50, for ages
5-9 the value of 440, and for the ages 10-14 the value of 1,210.
The thus reconstructed histograph (shown in the above tablej
may now be fitted to a logarithmically transformed Gram or
Charlier curve in the usual routine manner. The computations
of the relative moments w,. result in the following values, for
a provisional origin at the age of 17 and a unit interval of 5 years
?Wi = 1.8380, m, = 8.6342, ms =39.8630
From which we find
;.i = 1.838, X.2 = 5.2560, A.3=4.5824. (Mean at age 26.19.)
The equation
• ^r''-3?.2V = V
then becomes
4.582-/:-' -82.876>:' = 145.200,
therealrootof which is r = 19.0 (on basis of a 5 year unit.)
254
TRANSFORMED FREQUENCY CURVES
[129
We furthermore find that ?? =0.120 and w= 2.9227+%, 5 =
4.5321, which finally brings about the transformation of the
variate x by means of the formula
z = [loge(a:+68.8) -4.532] :0.12
where x is expressed in unit intervals of 1 year.
The further determination of the coefficients ^n, h and k^ by
means of the method of least squares results in the values:
fco = 947.4, A:;, = -63.4 and A:4=-30.0. Multiplying these
values with their respective values oifu(z), <f:i(z) and ip-iiz) and
forming the corresponding sums we finally obtain the second
component curve.
Sooo
\
zaoo
/
\
s*oo
/
\
ZAoo
1
\
azoo
1
\
t
\
Z ooo
1
\
1 fioo
fa>oo
/
\
/
/
l>*oo
y
r
\
I zoo
/
\
/
\
looo
J
7
\
>
^
)
— -_
-^
\
— -
■--_ L/
V,
"^T '~"~~~
— -
I
\
s^
^
\5 2o as 3o 3S >io .^s So ss <&o <&s 7o 75 ao ss -30 05 loo ,«s.<s,e-:s>.
FIGURE 3
Diagram showing Rraduation of <\s column in tiie .1 .l/('') table by a compound frequency curve of the
(irain-Cluirlicr ty[K'.s.
The sum of h\[x) and Fuix) as shown on page 255 and also
in the figure gives us the final compound frequency curve or
the d^ curve, from which it now is a simple matter to form the
l^ or irfj column and its co-ordinated column of r/^..
Graduation of Ameiiican Male Mortality Table (AM(5)) by means
OF a Compound Frequency Curve
Age
FKz)
Fii(x)
dx
h
lOOOgr
15
21
302
323
100532
3.21
6
26
319
345
100209
3.44
7
30
332
362
998G-4
3.62
129] MORTALITY TABLES 255
Age
F/d)
Fri{x)
(is
z;
lOOOg^
8
35
342
377
99502
3.79
9
39
350
389
99125
3.92
20
44
354
398
98736
4.03
1
48
354
402
98338
4.11
2-
57
352
409
97936
4.18
3
60
349
409
97527
4.19
4
68
343
411
97118
4.23
25
75
336
411
96707
4.25
16
86
327
412
96296
4.28
7
95
317
413
95884
4.31
8
106
308
414
95471
4.33
9
117
297
414
95057
4.36
30
130
285
415
94643
4.38
1
144
275
419
94228
4.45
2
IGO
261
421
93809
4.49
3
179
249
428
93388
4.58
4
198
238
436
92960
4.69
35
220
226
446
92524
4.82
6
242
213
455
92078
4.94
7
2G8
201
469
91633
5.12
8
296
188
484
91154
5.31
9
323
175
501
90070
5.53
40
360
164
524
90169
5.81
1
396
152
548
89645
6.11
2
436
141
577
S9097
6.47
3
478
128
606
8S520
6.80
4
526
117
643
87914
7.32
45
570
107
677
87271
7.76
6
630
96
726
86594
8.39
7
690
87
777
85868
9.05
8
754
78 ■
832
85091
9.7S
9
820
69
889
84259
10.55
50
893
61
954
83370
11.44
1
967
53
1020
82416
12.37
2
lor.i
47
1098
81396
13.49
3
1138
41
1179
80298
14.6S
4
1231
36
1267
79119
16.01
55
1324
30
1354
77852
17.39
6
1425
25
1450
76498
18.95
7
1526
52
1548
75048
20.62
8
1636
18
1654
73500
22.41
9
1745
15
1760
71846
24.54
60
1856
12
1868
70086
26.65
1
1970
11
1981
682 IS
29.04
2
2085
9
2092
66237
31.59
3
2201
7
2208
64145
34.42
4
2312
6
2318
61937
37.41
65
2420
5
2425
59619
40.67
256 TRANSFORMED FREQUENCY CURVES fl29
Age
F/{i)
Fii{x)
dT
Ix
lOOOgi
66
2522
4
2526
57194
43.62
7
2020
3
2623
54668
47.97
8
2700
2
2708
52045
52.02
9
2779
2
2781
49337
56.37
70
2848
1
2849
46556
61.19
1
2896
2896
43707
66.25
2
2922
2922
40811
71.60
3
2937
2937
37889
77.51
4
2930
2930
[34952
83.93
75
2898
2898
32022
90.51
6
2848
2848
29124
97.80
7
2772
2772
26276
105.48
8
2662
2662
23504
113.53
9
2553
2553
20836
122.50
80
2412
2412
18283
131.95
1
2251
2251
15871
141.84
2
2080
2080
13620
152.76
3
1895
1895
11540
164.21
4
1709
1709
9645
177.19
85
1504
1504
7936
189.52
6
1311
1311
6432
203.82
7
1125
1125
5121
219.68
8
947
947
3996
236.99
9
775
775
3049
254.18
90
624
624
2274
274.41
1
489
489
1650
296.36
2
374
374
1161
322.14
3
264
264
787
335.45
4
197
197
523
376.67
95
135
135
326
414.11
6
89
89
191
465.97
7
49
49
102
480.39
8
29
29
53
547.16
9
18
18
24
780.00
100
6
6
6
1000.00
It will be of interest to compare these latter values with the
original values of q^ as derived by Mr. Henderson's graduation.
Such a comparison is shown in the appended table for quin-
quennial ages.
Ages lleudorson's gx Fisher's g-i
15 3.46 3.21
20 3.92 4.03
25 4.31 4.25
30 4.46 4.38
35 4.78 4.82
130] ADDITIONAL EXAMPLES 257
Ages Henderson's qx Fisher's gx
40 5.84 5.81
45 7.94 7.76
50 11.58 11.44
55 17.47 ■ 17.39
60 26.68 26.65
65 40.66 40.67
70 61.47 61.19
75 91.94 90.51
80 135.74 131.95
85 197.07 189.52
90 280.35 274.41
95 387.76 414.11
100 562.50 1000.00
I think that every unbiased critic will admit that there exists
a satisfactory agreement between the two tables in spite of the
fact that we have worked throughout with basic data in 5-year
age groups. Moreover, the actual arithmetical work in the
case of the gi'aduation by means of compounded Gram or
Charlier curves is much simpler than the usual methods of
graduation by Makeham's formula and mechanical inteipola-
tion formulas as employed by Mr. Henderson.* Another point
speaking in favor of the frequency curve graduation is that our
resulting functions are continuous functions for which standard
tables of definite integrals have been prepared. It is therefore
possible to use the elegant and continuous method originally
introduced by Mr. Woolhouse in the computation of premiums
and policy values. Unfortunately this is not the place to treat
this interesting phase of the question, although we may in pass-
ing it mention that a gi-aduation of the kind as here presented
in practical computations of policy values and premiums is
even easier to work with than the renowned graduation formula
by Makeham, especially in the case of life contingencies involv-
ing 2 or more lives.
130. Additional Examples. — As another illustration I pre-
sent the following frequency distribution (arranged in groups of
3-year intervals) of the ages of a group of 19,274 male em-
ployees of the Bell System of the American Telephone and
*I do not wish to imply those remarks as a criticism of the able graduation by
Henderson, however.
258
TRANSFORMED FREQUENCY CURVES
[130
Telegraph Company, which most kindly has been furnished
to me through the courtesy of this company.
Age Distribution of Male Employees in the Bell System
Ages
X
F(x)
Ages
X
nx)
13-15
1
46-48
11
380
16-18
1
9
49-51
12
272
19-21
2
745
52-54
13
186
22-24
3
2,264
55-57
14
141
25-27
4
3,828
58-60
15
110
28-30
5
3,801
61-63
16
72
31-33
6
2,711
64-66
17
43
34-36
7
1,918
67-69
18
17
37-39
8
1,339
70-72
19
14
40-42
9
884
73-75
20
3
43^5
10
533
76-78
21
2
Choosing the provisional lower limit at age 14 we find the
following values for the crude moments or power sums s.
So = 19,274, 81 = 112,363, s. =794,771, §3=6,790,761
The values of the semi-invariants are
Xi = 5.830, Xo = 7.2478, X3=27.4191.
• The resulting cubic expansion is therefore
27.419>7''-157.592>72= 380.731
for which the solution is >7 = 6.1185. .
We have furthermore
6.1185 =6™'^ ■•"'"'
7.2478 = e-'"'-'"'(e"'-l), or
W' = 0.1768, w = 0.4205 and ?w = 1.5462
On the basis of an interval of one year we have therefore:
z = [log(x) - 13.1) -2.6451 :0.421*
as the value of the variate in the generating function (^0(2;).
We have Tn= 1.54G2 + log 3 = 2.645.
130]
ADDITIONAL EXAMPLES
259
The values of ko, h and ki as determined by the method of
least squares are A:o = 3064.4, k3=A5.1, A;4 = 80.5, on basis of one
year interval.
A comparison between the calculated and observed values
(the latter being shown by single ages) is given in the attached
diagram, which evidently is satisfactory for all practical pur-
poses. I wish here to mention that an attempt by some of the
FIGURE 4
Diagram showing comparison between observed and theoretical frequency distribution of active
group of male employees of the Bell System.
statistical assistants of the A. T. & T. Co. to fit the above data
by means of the Pearsonian curves proved futile. Personally
I have not as yet made an attempt to verify this negative
result.
As a final illustration we quote from J0rgensen's monograph
an application of the logarithmic transformaion of the pre-
viously discussed observations on the number of petal flowers
in Ranunculus Bulbosus. Since the variate in this instance is
integral, the obsei-\' ations themselves clearly indicate that there
must be a lower limit, or biological zero so to speak, at 4 petal
flowers.
The crude moments are then
5o = 222, 5i = 362, 5-2 = 794, from which we obtain
260 TRANSFORMED FREQUENCY CURVES [130
m =0.0440
n =0.5445
ko= 183.2,
so that the formula reads
1 SQ O 1 Hog (X— 4)— .044-12
^(^)._l^,-2i 0.5445 J
.5445 V 27r
The detailed calculations according to this formula are shown
below :
(1) (2) (3) (4) (5)
lognat(x-4) log(x-4)-m (2):n <Po(2) Fix)
.0000 - .0440 - .0810 .3989 131.8
.6932 + .6492 +1.1926 .1965 66.1
1.0986 1.0546 1.9373 .0612 20.6
1.3863 1.3423 2.4658 .0191 6.4
1.6094 1.5654 2.8756 .0064 2.2
1.7918 1.7478 3.2107 .0024 0.8
A closer fit could of course be had by adding additional terms
to the series, but even with one term the agreement between
calculated and observed values is quite satisfactory for all
practical purposes.
CHAPTER XVII
Frequency Curves and Their Relation to the Bernoullian Series
131. The Bernoullian Series. — In Chapter IX it was
shown that the general term
in the point binomial ip-{-Qy, where p is the a priori probability
for the happening of an event £" in a single trial, represents
the probability that E will happen x times and the comple-
mentary event, E, s — x times. We also found that the maxi-
mum term in the Bernoullian expansion of the point binomial
could be written as:
1
T„^ = —=
V 2Trspq.
when s is a large number.
We wish now also to find a more simple expression for the
general term, <p(x), instead of the laborious expression involving
factorials of high order.
It is evident that (fix) represents a frequency function of an
integral variate x which can assume all positive integral values
from to s, and which satisfies the property of all frequency
curves that
Z^(:*^) = Z(^)pY-^=(p+9r
= 1
We may therefore write (f{x) in the form of a Gram-Charlier
frequency series as
(p(x) ='Xci<fi{x) for i = 0,S, 4, 5 . . .
This involves the computation of the semi-invariants \(x)
for r = 0,3, 4, 5
By the definition of the semi-invariants we have:
Soe ■ "■ 3(^(a:) =2X^(a:)e'"^,
•2QL
262 FREQUENCY CURVES AND BERNOULLIAN SERIES [131
where a; = 0, 1, 2, 3 s and l(f(x) = (p+qy = 1, or
Xjto I X2a>8 I X3C03 .
soe^ "' " " "=.^(0) + <p(l)e"+c(2)e-" + . . .+^(s)e"-
which for CO =0 reduces to So = (p-\-qY = 1.
Taking the logarithm on both sides of the above equation
we have
Now it is easily seen that
^i,/(co)=^~j or Dl/(a)) (pe'^+q)^spe^
from which we find
Dlf{o^)q = pe-[s-Dlf{o^)]
Dlmq = pe-[s-Dlf{o^) -Dlfic,)]
Dlmq = pe-[s-DU{o^) -2 D:,/(a;) -D^/(a;)]
I>'/(co)9 = pe'^[s-DL/(a;) -3 D:,/(co) -3 DiJ{<^) - DtKo^)]
where
d^lV. ' 2! ' 3!
Letting co = we have therefore successively
T^nrr \ rf"|^CO Xoco=^ ?..C0'
'uq = p{S-\i)
'f~2q = V{s-'h\-'K2)
\zq ^p{s- Ai - 2X2 - A3)
A49 = pis - Xi - 3X2 - 3X3 - X4)
131] THE BERNOULLIAN SERIES 263
or
11 = sp
X2 = spq = (t2
'k3=spq{q-p)
>v4=spg(l— G pq)
The generating function <fo{x) in the Gram-CharUer series
may hence be written as
fTV27r ^ 2Tr spq
while the coefficients c.j and d of the third and fourth derivatives
of f o(x) according to the formulas from paragraphs 113 and 114
take on the form
(-1)3 _
C3= — ^^spq{q-p) = -{yj spq) \q-p):V.
(-1)4
C4= — —spq{l-Qpq)={spq) ^{.1-Qpq)-A\
which serve as measures for skewness and excess.
Since p and q are proper fractions whose product never can
exceed \i it is readily seen that for large values of s both c?, and
Ci will become very small quantities unless either p or q should
be so small that the product sp (or sq) itself would be small
even when s is a large number, a case which we presently shall
discuss in detail.
Apart from this exception the expression for the frequency
function
F{x)=(fo{x)-\-C?iip3{x)-\-Ci(Pi{x)+ . . .
approaches therefore the normal probability curve of Laplace
whenever s is a large number.
When, on the other hand, s in the point binomial (p-^-qY is
not a large number both c.{ and d play an important role as the
necessary correction factors. (For p = q = y^ all the semi-
invariants of uneven order vanish.)
264 FREQUENCY CURVES AND BERNOULLIAN SERIES [131
The normal form of the point binomial, or ^o(a:), was already
established by Laplace who also worked out a more accurate
expression for skew binomials, which expression can be shown
to represent the two terms <f^>[x) and C:nf3{x)
As an illustration of the above formulas we shall now try to
express a few Bernoullian point binomials by means of a
Gram-Charlier series.
Let us for instance try to express (.05 + . 95)'"" by a Gram-
Charlier series. We have in this case the following values for
the parameters
Xi = 5.0, VXo = a=2.1795, 6-3= -0.0688, C4 = 0.00625
The substitution of these values in the Gram-Charlier series
results in the following relative frequency distribution:
X
^(.r)
X
Fix)
.0084
8
.0614
1
.0312
9
.0343
2
.0763
10
.0179
3
.1356
11
.0081
4
.1812
12
.0031
5
.1865
13
.()()()9
6
.1522
14
.0003
7
.1028
15
.0001
A similar calculation in the case of the Bernoullian binomic^i
r0.1+0.9;'"» gives
X, = 10, a = 3, C3 = - .0445, c^ = 0.0021
with the following distribution:
X
Fix)
X
Fix)
.0000
12
.0984
1
.0004
13
.0732
2
.0020
14
.0502
3
.0065
15
.0322
4
.0162
16
.0194
5
.03:i3
17
.0109
6
.0581
18
.0058
7
.0,S75
19
.0027
8
.1145
20
.0012
9
.1318
21
.0005
10
.1338
22
.0002
11
.1211
23
.OOOl
132; poisson's exponential 265
We shall presently have occasion to compare these distribu-
tions with those obtained from a direct expansion of the point
binomial.
132. Poisson's Exponential. The Law of Small Num-
bers. — In certain statistical series it frequently happens that
the semi-invariants of higher order than zero all are equal, or
that
Ai = Ao = A3 = . . • . . = A^ =^ A.
We shall for the present limit our discussion to homograde
series where the variate is always positive and integral, and
where therefore the definition of the semi-invariants is of the
form:.
= ^(0)eO-+^.(l)ei-+^.(2)e--+^(3)e^''^+ . . . , or
e^' "' '^' =e-V'''" = iV(a:)e""for x = 0, 1,2,3 ... ,
which also can be written as
(Xp^ A2^2w \
l+-Y7+^r+ ; =f(0)l+v^(l)e'^+c(2)/'^+
The coefficient of e'"'^ gives the relative frequency or the
probability for the occurrence of x = r, and we henceforth find
that
(r(x)=xp(r)= j—
This is the famous Poisson Exponential, so called after the
French mathematician, Poisson, who first derived this expres-
sion in his Recherches sur la Prohabilite des jugesments, but in
an entirely different manner than the one we have indicated
above.
The Poisson Exponential opens now a way to the treatment
of the point binomial in the exceptional cases where the product
sp (or sq) is small even when s is a very large number, or when
more strictly speaking the expression
lim sp = X
where ?» is a finite number.
266 FREQUENCY CURVES AND BERNOULLI AN SERIES [132
Under such conditions p (or q) must approach zero and its
complementary probability q (or p) must approach unity as
their limiting values.
The expressions for the semi-invariants as given in paragraph
131, i. e.
^2 = spq
^3 = spq(q-p)
?.i=spq{l-6pq)
will under these conditions all approach the limit sp, and the
general term in the Bernoullian expansion of the point binomial
can therefore be expressed by means of the Poisson exponen-
tial.
In all cases where the semi-invariants of various orders hap-
pen to be equal, or very nearly equal, the formula by Poisson
will be preferable in place of the more general expansion by the
Gram-Charlier series.
As an illustration we may select the simple binomial (.001 + .
.999) ""^ where the semi-invariants have the following values:
?.i=0.1, Xo = 0.0999, As = 0.099702, X, = 0.0994006,
and therefore may be considered as being nearly equal.
The general term, (f{x), in this particular point binomial can
therefore be written as a Poisson exponential of the form:
,^(x)=,/.(r)=e-o-^0.1^:r! for r = 0, 1,2,3 ....
The Russian statistician, Bortkewitsch, has given in his in-
teresting and scholarly brochure Das Gesetz der kleinen Zahlen
(1898) a four decimal place table of the Poisson exponential
e~^ /l'':r ! for values of X from 0,1 to 10.0. The English biometri-
cian, Soper, in 1914 published a 6 decimal place table from
X = 0.1 to z*^ = 15.0. This table is found in Pearson's well-known
Tables for Biometricians. For the above mentioned Bernoul-
lian point binomial (0.0014-0.999)^'"', corresponding to the
Poisson exponential e""" 'O.l'ir!, we find from Soper's table the
following values of \l/{r).
132] poisson's exponential 267
r
xjjir)
.904837
1
.090484
2
.004524
3
.000151
4
.000004
While the exponential of Poisson requires theoretically at
least, that the semi-invariants must all be of the same magni-
tude, it will, however, often be found that this exponential will
give a fair approximation to the true observed values of the
frequency curve in cases where the semi-invariants Xi, X2, X3,
X4 . . .do not differ greatly from each other. In this connec-
tion it is of interest to compare the fits of Poisson's exponential
and the Gram-Charlier series with the true values in the binom-
ial expansion in the three examples we have given above.
Through the courteous efforts of my translator and co-editor.
Miss C. Dickson, the three point binomials (0.001+0.999)'°'',
(0.05+0.95)1°'' and (0.10+0.90)""' have been expanded directly
and the results as compared with the forms of Poisson and of
Gram-Charlier are shown in the following tables:
Values of <fix) ix Various Point Binomials
TABLE I (0.001+0.999)100
X
Binomial
Poisson
.9048
.9048
1
.0906
.0905
2
.0045
.0045
3
.0001
.0002
4
.0000
.0000
TABLE II (0.05+0
.95)100
X
Binomial
Gram-Charlier
Poisson
.0059
.0084
.0067
1
.0312
.0312
.0337
2
.0812
.0763
.0842
3
.1396
.1356
.1404
4
.1781
.1812
.1755
5
.1800
.1865
.1755
6
.1500
.1522
.1462
7
.10(50
.1028
.1044
8
.0()49
.0614
.0653
9
.0349
.0343
.0363
10
11
12
.0167
.0072
.0028
.0179
.0081
.0031
.0181
.0082
.0034
13
.0003
.0009
.0013
14
.0001
.0001
.0005
268 FREQUENCY CURVES AND BERNOULLIAN SERIES [132
TABLE ill (0.1+0.9)100
X
Binomial
Gram-Charlier
Poisson
.0001
.0000
.0000
1
.0003
.0004
.0005
2
.0016
.0020
.0023
3
.0059
.0065
.0076
4
.0159
.0162
.0189
5
.0339
.0333
.0378
6
.0596
.0581
.0630
7
.0889
.0875
.0901
8
.1148
.1145
.1125
9
.1304
.1318
.1251
10
.1319
.1339
.1251
11
.1199
.1211
.1137
12
.0988
.0984
.0948
13
.0743
.0732
.0729
14
.0513
.0502
.0521
15
.0327
.0322
.0347
16
.0193
.0194
.0217
17
.0106
.0109
.0128
18
.0054
.0058
.0071
19
.0026
.0027
.0037
20
.0012
.0012
.0019
21
.0005
.0005
.0009
22
.0002
.0002
.0004
23
.0000
.0000
.0002
These tables need no further explanation and demonstrate
the great graduating ability of the Posison function in the
case of point binomials. For although the Gram-Charlier
functions give better results in the last example, we must on
the other hand not forget that we had to determine 4 para-
meters, X,, a, C3 and d, while Poisson's exponential requires only
the determination of the single parameter X. The great draw-
back of the Poisson exponential lies in the fact that it is a dis-
crete function, which exists only for positive integral values of
the variate. It therefore does not lend itself so readily to in-
tegration as the Laplacean probability function.
It is the great achievement of the eminent Swedish astrono-
mer and statisitican, Charlier, to have been the first to attempt
to find a continuous function possessing the same powerful and
flexible characteristics as the Poisson exponential. Charlier
has introduced as a generating function a certain curve type
which is expressed by the following formula:
133] THE LAW OF SMALL NUMBERS 269
TT
\P(x) =4 m,n(^) = / e'^^'^^cos [nsino)—xo)]do^
o
Using the above expression as a generating function Charlier
has shown that any frequency curve can be expressed by the
series
F(x)=V^(x)-^-A^(x)+|jA^i//(x)-^jAV(x)+
At the time when Charlier introduced this function in the
theory of frequency curves in a little pamphlet Weiteres uher das
Fehlergesetz he gave only a rather short sketch of the method.
He and the Danish actuary, J0rgensen, have, however, of late
years further developed the method so as to make it useful in
practical computations. J0rgensen has in this respect done some
very neat work in supplying a method for determining the con-
stants, k, in the above series by means of semi-invariants
We intend to treat these investigations in the forthcoming
second volume of this treatise. In the meantime, however,
students who encounter skew frequency distributions in their
work will in nearly all cases be able to overcome the practical
difficulties of a graduation by means of the logarithmic trans-
formation of the variate, treated in the earlier chapters of this
book.
133. The Law of Small Numbers. — In the case of integral
variates we wish, however, to call the reader's attention to cer-
tain properties of the Poisson exponential, or probability func-
tion, which have been apparently overlooked or misunderstood
by several writers, especially among the English biometricians.
Somehow or other the impression among those writers has been
that the frequency curve or probability function of Poisson is
invariably connected with the expansion of the point binomial.
Thus for instance Mr. Yule in his well-known Introduction to
the Theory of Statistics treats it as such, while Lucy Whitaker
in an article entitled "On the Poisson Law of Small Numbers"
in Biometrika for April, 1914, subjects the whole theorem to
a scathing criticism. We quote the following sentence from
Miss Whitaker 's article: 'Tt might be supposed, although
erroneously, that the Poisson-Exponential formula was capable
270 FREQUENCY CURVES AND BERNOULLIAN SERIES [133
of great accuracy in addition to its great simplicity. But this
is to neglect the fundamental assumptions on which it is based,
namely :
(1) That the data actually correspond to a binomial
(2) That in the binomial q is small and n large."
It is true that Poisson in deriving his formula started from
the BernouUian point binomial so as to meet the cases where
the normal probability curve of Laplace failed to give a close
approximation, that is in the case of the limiting value when
either p or q becomes very small, but s is large enough so that
sp or sq remain finite; and it is probably this fact which has
prompted the above remarks of Miss Whitaker. But this is
really to put the cart before the horse. In the case of integral
variates we can, as shown in the preceeding paragraphs, derive
the Poisson probability function as a general form of frequency
distributions whose semi-invariants are all equal, and it is only
incidental that this property is possessed by the special binomial
limit in the case mentioned above. But this property is not a
general property of the binomial any more than it is the property
of the same point binomial function to result in a Laplacean
normal probability curve when the exponent s is large and
Looked at wholly from the point of view of frequency func-
tions it is, however, not necessary to resort to the binomial
limit as a base for the derivation of Poisson's formula, which
can be derived directly from the definition of the semi-invariants
when those peculiar parameters are considered as being of
equal magnitude irrespective of their order. Now the question
presents itself whether the Poisson probability curve might
not, like its fellow brother, the Laplacean probability curve,
be used as a generating function in expanding certain types of
frequency distributions in serial form. It is to this question
that the discussion is devoted in the following chapter.
CHAPTER XVIII
POISSON-CHARLIER FREQUENCY CURVES FOR INTEGRAL VARIATES
134. Charlier's B Curve. — We have already seen in the pre-
vious chapters that the Gram-Charlier frequency curve could be
written as
F (x) = :i.Ci<Pi(x) =^CiHi{x)<f oix) for i = 0, 1, 2, 3, ... .
where (po{x) is the generating Laplacean probability function.
The idea now immediately suggests itself to use a similar
method of expansion in the case of the Poisson probability
function and to employ this exponential as a generating func-
tion in the same manner as the Laplacean function. We are,
however, in the present case of the Poisson exponential dealing
with a generating function which so far has been defined for
positive integi'al values only and, therefore, represents a dis-
crete function. For this reason it will be impossible to express
the series as the sum-products of the successive derivatives of
the generating function and their correlated parameters c. We
can, however, in the case of integral variates express the series
by means of finite differences and write F{x) as follows:
Fix) =coxjj(x) -\-CiAyjj{x) -^Co^-xfJ(x) + .... (/)
where xpix) =e~''^irf:x! for a: = 0, 1, 2, 3, .... , and
^"i//(x)=l,
Ayp{x)=yp{x)-Hx-l),
A'^Pix)=Arp(x)-AHx-l) =4^{x) -2\l^(x-l) -{-4^(x-2).
The series (/) is 'known as the Poisson -Charlier frequency
series or Charlier's B type of frequency curves.
The semi-invariants of these frequency series are given by the
following relation:
e "^ 3! = J^Mix) +CiA'/'(a:)-hC2AV(x) + ] e^"
X =o
271
272 POISSON-CHARLIER FREQUENCY CURVES [134
Expanding and equating the co-efficients of equal powers
of CO we have :
Xo = 1 = co^4'(x) or Co = 1
Xi = :ix {^(x) -hcMU) +C2AV(a:) + ) UD
?ix'-\-?.2 = ^JcKxlf(x) -\-c,Axjj{x) -\-C2A-xIj{x) -{- )
We now have
X\p{x) =1, and
Ixxjfix) ='^me~"'m''~^ :(x — 1)1 = 7n^xfj(x — 1) =m.
We also find from well-known formulas of the calculus of
finite differences that*
^x^{x) =m^+m
'XxA\jf{x) = — 1
Xx^Axjj(x) = -(2m-\-l)
Xx^A^(x)=2
Substituting these values in (//) we obtain
Pui-+X2 = m-+m — (2?w+l)ci+2c2
By letting in = Ai we can make the coefficient Ci vanish, which
results in
Ci = M[/^2 — m]
where the two semi-invariants Xi and A2 are calculated around
the natural zero of the number scale as origin.
For the above discussion we have limited ourselves to the
determination of the three constants m, Co and C2. It is easy,
however, to find the higher parameters Cs, Ci, Cs, . . . . from
the relations between the moments of the Poisson function and
the semi-invariants of order 3, 4, 5, . . . etc. Charlier usually
*Th«'s(; formulas ran also h(? dorivcfl from the definition of tho semi-invariants and
tho well-known relations between moments and semi-invariants as niven on page 237,
when we remember that according to our dclinition all semi-invariants in the Poisson
exponential are »'<(ual to m.
135]
NUMERICAL EXAMPLES
273
calls the parameter m the modulus and C2 the eccentricity of the
B curve.
135. Numerical Examples. — As an illustration of the appli-
cation of the Poisson-Charlier series we select the previously-
mentioned series of observations on alpha particles radiated from
a bar of Polonium as determined by Rutherford and Geiger
and shown on page 174 of this treatise. We are here dealing
with integral variates which can assume positive values only
and the observations are therefore eminently adaptable to the
treatment by Poisson-Charlier curves. Selecting the natural
zero as the origin of the co-ordinate system we find that the
first two semi-invariants are of the form
?.i =3.8754, Ao = 3.6257, and we therefore have:
w = >.i=3.88; C2 = H[>.2-w] = -0.125
The equation for the frequency distribution of the total N =
2608 elements therefore becomes
F(x)=Ml//3.88(x) + ( -0.125) A-1//3.88(X)].
The table below gives the values as fitted to the curve, F{x) :
Alpha Particles Discharged from Film of Polonium
(Rutherford and Geiger)
A' =2608, m = 3.88,
C2= -0.125
(1)
(2)
(3)
(4)
(5)
(0)
X
-^Cx)
A2'i(x)
.VX(2)
iVx(3)Xc3
(4) + (5)
.020668
+.020668
.53.9
- 6.7
47
1
.080156
+.038820
209.0
-12.7
196
2
. 1 5.5455
+.015S11
405.4
- 5.2
400
3
.201015
-.029739
524.2
+ 9.7
533
4
.194967
-.051608
508-5
+ 16.8
525
5
.151265
- .037()54
.394.5
+ 12.3
407
6
.097850
-.009714
254.9
+ 3.2
258
7
.0.^4249
+ .009814
141.2
- 3.2
138
8
.026316
+.015668
68.7
- 5.1
64
9
.011351
+.012968
29.6
- 4.2
25
10
.004407
+.00,S021
11.5
- 2.6
9
11
.001555
+ .004092
4.1
- 1.2
3
12
.000503
+.001S00
1.3
- 0.6
1
13
.000150
+.000699
0.4
- 0.2
14
.000042
+.000245
0.1
- 0.1
15
.000010
+ .000076
0.0
- 0.0
16
.000003
+ .000025
17
.000001
+.000005
274 POISSON-CHARLIER FREQUENCY CURVES 136
Bateman has in Philosophical Transactions (1902) given a
theoretical frequency distribution of the above series of observa-
tions wherein he develops the Poisson probability function,
being ignorant of the previous demonstration by Poisson. In
a later note he mentions that the formula was given by the
French mathematician in his work on probabilities, published
in 1837.
Bateman's calculation includes, however, only the first term
of the Poisson-Charlier series and is, therefore not so close as
the above fit.
As a second example we offer our old friend, the distribution
of fiower petals in Ranunculus Bulbosiis. Selecting the zero
point at a: = 5 and computing the semi-invariants in the usual
manner we obtain the following equation for the frequency
curve.
F{x) =222i//(x)+31.5A-i//(a;), ?w =0.631
A comparison between calculated and observed values fol-
lows below:
X
F(x)
Obs.
5
134.9
133
6
51.6
55
7
22.5
23
8
9.5
7
9
2.9
2
[0
0.6
2
136. Transformation of the Variate. — For integral vari-
ates we have shown that the Poisson frequency curve possesses
the important property that all its semi-invariants are equal.
Now while a frequency distribution of a certain integral variate,
X, may perhaps 7wt possess this property, it may, however,
very well happen after a suitable linear transformation has been
made, that the variate thus transformed will be subject to the
laws of Poisson's function.
Let z = ax — b represent the linear transformation which is
subject to the above laws with a series of semi-invariants all
equal to m.
These semi-invariants according to the properties set forth
in paragraph (104) are therefore
136] TRANSFORMATION OF THE VARIATE 275
m = Xi (2) = aXi (x) — 6
m = X3(z) ^a^Xiix)
and our problem is to find the unknown parameters a, b and m.
Simple algebraic methods, which it will not be necessary to
dwell upon, give the following results:
a = Xo :?.s
As a numerical illustration of this transformation we choose
from J0rgensen a series of observations by Davenport on the
frequency distribution of glands in the right foreleg of 2000
female swine.
No. of Glands 0123456789 10
Frequency . . 15 209 365 482 414 277 134 72 22 8 2
The values of the three first semi-invariants are
Xi =8.501, P.o = 2.825, X3 =2.417,
a = 2.825:2.417 = 1.168
m = 2.825-^:2.417- = 3.859
b = (1.168) (3.501) -3.859 =0.230.
The new variable then becomes z=ax — b and the transformed
Poisson probability function takes on the form:
... e-"'m'
In general, however, we will find that z is not a whole number
and the expression z\ therefore has no meaning from the point
of view of factorials at least. This difficulty may, however,
be overcome through the introduction of the well-known
Gamma Function, 1X2; + 1), which holds true for any positive
276 POISSON-CHARLIER FREQUENCY CURVES [137
or negative real value of z and which in the case of integral
values of z reduces to V{z^\)=z\
Hence we can write the transformed Poisson probability
function as
xfj{z)
r(2+i)
Tables to 7 decimal places of the Gamma Function, or rather
for the expression —log Fl^ + l), have been computed by J0r-
gensen in his aforementioned book from z= —3 to z = 15, pro-
gressing by intervals of 0.01.
By means of this table and the tables of ordinary logarithms
it is now easy to find the values of i/>(z) in the case of the example
relating to the number of glands in female swine. The detailed
computation is shown below.*
(1)
(2)
(■■i)
(4)
(o)
(6)
7
X
z —
AogViz+l)
log m^
(3) + (4)+log<r-'«
^{z)
Fix)
-.230
.9209
.8651
.1101-2
.0129
30.1
1
+.938
.0108
.5500
.8849-2
.0767
179.2
2
2.106
.6555
.2350
.2146-1
.1639
382.9
3 -
3.274
.0679
.9199
.3119-1
.2051
479.1
4
4.442
.3216
.6048
.2.505-1
.1780
415.8
5
5.610
.4547
.2897
.0685-1
.1171
273.6
6
6.778
.4904
.9746
.7891-2
.0615
143.7
7
7.946
.4446
.6595
.4282-2
.0268
62.6
8
9.114
.3285
.3444
.9970-3
.0099
23.1
9
10.282
.1506
.0294
.5041-3
.0032
7.5
10
11.4r)0
.9177
.7143
.9561-4
.0009
2.1
137. The Bernoullian Series Expressed by B Curves.
In the case of the Bernoullian point binomial we have ?-i = sp and
'Ao^spq. If we now wish to express the general term, c{x), in
the binomial by a Poisson-Charlier series we evidently have
f{x)=\fj(x)-\-C2A-^{x) where
xIj{x) =e~"'m':xl
Now m = ?-! =sp, and Co = l^iy^-o—m) = Vzispq—sp) = — ^sp^
As an illustration of this expansion we may again look at the
*Tlie characteristics of the logarithms have been omitted in this talilo (except In
cohimn .5) and only the positive mantissas are shown. Column 7 represents tlio 2000
individual observations pro rated according to column 6.
138] REMARKS ON MR. KEYNES' CRITICISM 277
point binomial (0.1+0.9)'"", discussed on page (268). There
we have:
w = 10, C2 = — ^2> while
<f{x) =i//io(x) - 3/^A-i//io(x).
The actual computation by means of this formula results in
the following tabular representation.
X
Poisson-Charlier
X
Poisson-Charlier
.0000
12
.0986
1
.0002
13
.0744
2
.OOIG
14
.0514
3
.0059
15
.0330
4
.0159
16
.0197
5
.0340
17
.0108
6
.0599
18
.0055
7
.0893
19
.0025
8
.1149
20
.0011
9
.1301
21
.0005
10
.1314
22
.0001
11
.1194
23
.0000
A comparison of this series with the actual expansion of the
point binomial by Miss Dickson on page (268) leaves little to
be desired in the way of exactitude. The fit to the true binomial
is even closer than that of Gram-Charlier series in spite of the
fact that only two parameters enter into the determination of
Poisson-Charlier's curves while four parameters are required
for the Gram-Charlier curves.
I*i8. Remarks on Mr. Keynes' Criticisms. — From the above dis-
eussiou it is (.-videut that the BeruouUiun poiut binomial can always be
represented without difficulty by either the Gram-Charlier or the Poisson —
(^harlier frequency curves. This point is of interest in connection with
some wholly misleading and erroneous statements regarding Laplace's
analysis of the Bernoullian Theorem by ZVIr. .J. M. Keynes, in his recently
issued ''Trcaliae on Prohahility.
On page 358 JNIr. Keynes points out the ass\Tnmetry in the Bernoullian
expansion and claims that the want of symmetry is generally Iteing over-
look(!d, "and it is not uncommon to assume that the probability of a given
divergence less than pm is equal to that of the same divergence in excess
of -pm, and, in general, that the probability of the frequency's exceeding
-pm in a set of m trials is equal to that of falling short of pm."
No real mathematician, and least of all Laplace, has ever claimed the
presence of symmetry as being general in the case of the Bernoullian
278 POISSON-CHARLIER FREQUENCY CURVES [138
series. Those who ha\ e fulUu into that error are economists and statisti-
cians who like ]Mr. Keynes are ignorant of the true assumptions underlying
Bernoulli's and Laplace's demonstrations. Every mathematician knows
for instance that the annual loss ratios for total permanent disability bene-
fits in workmen's compensation assurance on the basis of a small sample
pajToll of say 100,000 Ki'oner can assume all possible real values from
zero and upwards, and losses in excess of 100,000 are indeed possible.
Statistics fi-om certain Scandinavian industries show that the average
annual loss ratio in respect to total permanent disability is about 1 Krone
for each 100,000 Kroner of payroll exposure. We know, however, of a
certain instance in the case of a small industrial establishment with a
pajToU of nearly 100,000 Kroner where the losses in a single year were
more than 20,000 Kroner. In the great majority of cases, however, the
annual losses are nil. We are, therefore, dealing with a decidedly skew
Poisson — Charlier frequency curve.
Mr. Keynes' example is in fact less striking. He considers the case of
throwing aces in 00 successive throws with a die, and he remarks that the
ace cannot appear less often than not at all, whereas it may well appear
more than 20 times. This, of course, is self evident and realized by
Laplace in his analysis, and there is no valid reason for dw^elling at length
on such simple matters. But the English scholar is evidently greatly
impressed by these very simple considerations for he continues in a most
charming and naive manner as follows:
"The actual measurement of this w-ant of symmetry and the determina-
tion of the conditions, in which it can be safely neglected, in\ olves labori-
ous mathematics of which I am only acquainted with one direct investiga-
tion, that published in the Proceedings of the London MaUieinalical Society
by Mr. T. C. Simmons."
How this eharmi^ng and naive statement reminds one of the playful
sophistries of a bright and impish, but not necessarily bad, small boy,
trying to offer some excuse and explanation for his mischievous pranks,
while he at the same time is wholly unmindful of the fact that his explana-
tions and excuses are the most damning evidence of his own guilt.
Here we have Mr. Keynes, a succ(>ssful wTitcr of et^onomic sul)j(H't s, posing
as a critic of such intellectual giants in the realm of mathematical science
as Bernoulli, Laplace and Poisson (which of course presupposes that he
must have read vi^ry carefully the various writings of those old masters) ;
who calmly and in the most innocent manner admits that of the "labori-
ous" mathematics involved in this question he is only acquainted with a
rather clumsy demonstration publislicd in 180().
While I have not the slightest doubt as to the veracity of these facts
so far as Mr. Keynos is concerned, it will be of interest to see what the
actual historical facts are. Now in so far as the measurement of the
assymmetry or skewness of the BernouUian point binomial is concerned
this w'as already performed by Laplace himself, an ac(!omplishment which
in itself creates a degree of doubt in the reader's mind as to whether Mr.
Keynes really has stu(li(>d Laplace with the necessary care required of one
who poses as a critic of the great Freneluiuiu. Ilnrahl Westergaard, the
138] REMARKS ON MR. KEYNES' CRITICISM 279
eminent Danish scholar, whose fame as a statistician surely rests on a far
more secure foundation than that of Keynes takes in his Slatislikens Teori i
Grundrids (Copenhagen, 1915), special pains to j)oint out that Laplace
was the first to give a mathematical measure of the sktewness in a Ber-
noullian frequency distribution.
The Danish actuary. Gram, long before the intellect of our recent
English critic saw the light, derived his general series for frequency cm'ves,
which of course also applies to the Bernoullian case. Thiele in his Al-
mindelig lagttagclseslaerc, at a time when the young Keynes probably
was being piloted by his nursemaid or governess, discussed the same series
from the point of view of semi-invariants. Later on Charlier continued
in dii'ect line from where Laplace and Poisson concluded their labours.
The necessary corrections to the generating functions, whether these
be Laplace's or Poisson's probabilitj^ curves, as derived by these Scandi-
navian writers are given in Chapters XVII and XVIII of this treatise.
Not a single one of these demonstrations requires the "laborious" mathe-
matics as mentioned by Mr. Keynes. The fact that our romancing
English economist evidently is in blissful ignorance of the fundamental
w^ork by the Scandinavian school is, however, no excuse for his superfi-
cial knowledge of the expansion of statistical series, since much of this
work has appeared ^n English.
Mr. Keynes' misconception of the real significance of the Law of Small
Numbers and his criticism of Bortkewicz may possibly also be traced to
his apparent ignorance of the work of the Scandinavian authors. His
criticism moves practically along the same lines as that of the \iews held
by Miss Whitaker and described on page 270 of this treatise. He, like
Miss Whitaker, fails to realize that the generating Poisson probability
function arises from the more general fact that all its semi-invariants are
equal, rather than from the more special fact that one of the limiting values
of the point binomial reduces to a Poisson frequency curve. The very
fact that Mr. Keynes in his large volume never mentions the semi-invari-
ants leads one to inquire whether those important statistical parameters
remain a closed book to him. *
♦Similar immature views as those lield by Keynes and Miss Whitaker are also ex-
pressed by Mr. A. Mowbray in the May. 1920, Proceedings of the Casualty Actuarial
Societn ff America (page 107). .ludging from iiis tedious and laborious analysis Mow-
bray evidently never has heard of the semi-invariants.
280 ABRIDGED TABLE OF LAPLACE'S
z fo{z) fziz) fiiz) foiz) <f6iz) f<foiz)dz
0.00
.3989
+0.0000
+ 1.1968
-0.0000
-5.9841
.5000
.05
.3984
.0597
1.1894
.2983
5.9319
.5199
.10
.3970
.1187
1.1671
.5915
5.7763
.5398
.15
.3945
.1762
1.1304
.8743
5.5208
.5596
.20
.3910
.2315
1.0799
1.1420
5.1711
.5793
.25
.3867
.2840
1.0165
1.3900
4.7351
.5987
.30
.3814
.3330
.9413
1.6142
4.2223
.6179
.35
.3752
.3779
.8556
1.8111
3.6439
.6368
.40
.3683
.4184
.7607
1.9777
3.0122
.6554
.45
.3605
.4539
.6583
2.1117
2.3414
.6736
.50
.3520
.4841
.5501
2.2114
1.6448
.6915
.55
.3429
.5088
.4378
2.2760
.9371
.7088
.60
.3332
.5278
.3231
2.3052
- .2324
.7257
.65
.3230
.5411
.2078
2.2995
+ .4555
.7422
.70
.3123
.5486
+ .0937
2.2601
1.1135
.7580
.75
.3011
.5505
- .0176
2.1888
1.7298
.7734
.80
.2897
.5469
.1247
2.0880
2.2938
.7881
.85
.2780
.5381
.2260
1.9604
2.7964
.8023
.90
.2661
.5245
.3203
1.8095
3.2304
.8159
0.95
.2541
.5062
.4067
1.6387
3.5898
.8289
1.00
.2420
.4839
.4839
1.4518
3.8715
.8413
1.05
.2299
.4580
.5516
1.2529
4.0735
.8531
1.10
.2179
.4290
.6091
1.0458
4.1958
.8643
1.15
.2059
.3973
.6561
.8346
4.2403
.8749
1.20
.1942
.3635
.6925
.6230
4.2103
.8849
1.25
.1826
.3282
.7185
.4147
4.1107
.8944
1.30
.1714
.2918
.7341
.2130
3.9475
.9032
1.35
.1604
.2.549
.7399
- .0209
3.7278
.9115
1.40
.1497
.2180
.7364
+ .1590
3.4595
.9192
1.45
.1394
.1815
.7243
.3244
3.1510
.9265
1.50
.1295
.1457
.7042
.4735
2.8109
.9332
1.55
.1200
.1111
,6772
.6051
2.4481
.9394
1.60
.1109
.0780
.6440
.7181
2.0712
.9452
1.65
.1023
.0468
.6057
.8121
1.688()
.9505
1.70
.0940
+ .0175
.5632
.8870
1.3079
.9554
1.75
.0863
- .0094
.5173
.9431
.9363
.9599
1.80
.0790
.0341
.4692
.9809
.5801
.9641
1.85
.0720
.0.563
.4195
1.0014
+ .2450
.9678
1.90
.0656
.0760
.3693
1.0058
- .0646
.9713
1.95
.0596
.0933
.3192
.9955
.3452
.9744
PROBABILITY FUNCTION AND ITS DERIVATIVES 281
z
fo(2)
^3(2)
^4(2)
^5(2)
^6(2) f<fo{z)'h
2.00
.0540
-O.IOSO
-0.2700
+0.9718
-0..5!)39
.9772
2.05
.0488
.1203
.2223
.9366
.8091
.9798
2.10
.0440
.1302
.1765
.8915
.9899
.9821
2.15
.0396
.1380
.1332
.8382
1.1362
.9842
2.20
.0355
.1436
.0927
.7784
1.2488
.9861
2.25
.0317
.1473
.0554
.7139
1.3291
.9878
2.30
.0283
.1492
- .0214
.6460
1.3788
.9893
2.35
.0252
.1495
+ .0092
.5764
1.4004
.990()
2.40
.0224
.1483
.0362
.5064
1.3965
.9918
2.45
.0198
.1459
.0598
.4372
1.3701
.9929
2.50
.0175
.1424
.0800
.3697
1.3242
.9938
2.55
.0154
.1380
.0968
.3050
1.2619
.9946
2.60
.0136
.1328
.1105
.2438
1.1865
.9953
2.65
.0119
.1270
.1213
.1865
1.1007
.9960
2.70
.0104
.1207
.1293
.1338
1.0076
.9965
2.75
.0091
.1141
.1347
.0858
.9098
.9970
2.80
.0079
.1073
.1379
.0429
.8097
.9974
2.85
.0069
.1003
.1391
+ .0049
.7095
.9978
2.90
.0060
.0934
.1385
- .0281
.6110
.9981
2.95
.0051
.0865
.1364
.0563
.5159
.9984
3.00
.0044
.0798
.1330
.0798
.4255
.9987
3.05
.0038
.0732
.1284
.0989
.3407
.9989
3.10
.0033
.0669
.1231
.1140
.2624
.9990
3.15
.0028
.0609
.1171
.1253
.1911
.9992
3.20
.0024
.0552
.1106
.1332
.1271
.9993
3.25
.0020
.0499
.1039
.1381
.0705
.9994
3.30
.0017
.0449
.0969
.1404
- .0213
.9995
3.35
.0015
.0402
.0899
.1403
+ .0207
.9996
3.40
.0012
.0359
.0829
.1384
.0561
.9997
3.45
.0010
.0318
.0761
.1348
.0849
.9997
3.50
.0009
.0283
.0694
.1300
.1078
.9998
3.55
.0007
.0249
.0631
.1242
.12.-)4
.9998
3.60
.0006
.0219
.0.570
.1175
.1380
.9998
3.65
.0005
.0193
.0513
.1104
.1464
.9999
3.70
.0004
.0168
.0460
.1030
.1510
.9999
3.75
.0004
.0146
.0410
.0954
.1525
.9999
3.80
.0003
.0127
.0365
.0878
.1512
.9999
3.85
.0002
.0110
.0323
.0803
.1478
.9999
3.90
.0002
.0095
.0284
.0730
.1426
.99995
3.95
.0002
.0081
.0249
.0660
.1361
.99997
ADDENDA.
APPENDIX AND BIBLIOGRAPHICAL NOTES.
Chapter I.
Page 3. The establishment of the relations between hypothetical judg-
ments and probabilities is probably first due to F. C. Lange. See also the
discussion in Sigwart's "Logic" (English translation, Macmillan Co., New
York, 1904). A defense of the "principle of insufficient reason" as opposed
to the view of von Kries is given by K. 8titmpf ("tjber den Begriff der mate-
matischen Wahrscheinlichkeit") Ber. hayr. Ak. (phil. KL), 1892. For a
further discussion of the philosophical aspect the reader is advised to consult
"Theorie und Methoden der Statistik" (Tubingen, 1913) by the Russian
statistician, A. Kaufmann.
Chapter II.
Page 21. An interesting account of the application of the theory of proba-
bilities to whist is given by Poole in "Philosophy of Whist Play" (New York
and London, 1883). Page 23. Example 6. This is a general case of the
so-called game of "Treize" or "Recontre" first di.scu.ssed by Montmort in his
"Essai sur les Jeux des Hazards" (1708). "Thirteen cards numbered 1, 2,
3, ... up to 13 are thrown promiscuously into a bag and then drawn out
singly; required the chance that once at least the number on a card shall
coincide with the number expressing the order in which it is drawn." This is
one of the stock problems in probability and has been discussed by nearlv all
the leading classical writers on the subject.
Chapter IV.
The close connection between probability and symbolic logic is admirably
discussed by the Italian mathematician, Peano, in various of his mathematical
texts. Page 42. Example 19. See also the discussion by R. Hendersox in
"Mortality and Statistics" (New York, 1915).
Chapter V.
38. The moral expectation has been discu.ssed by Harald Westergaard
in "Tids.skrift for Matematik" (1878) and in " Smaaskrif ter tilegnede C. F.
Krieger" (Copenhagen, 1889).
Chapter VI.
A German translation with explanatory notes of Bayes's brochure has
recently appeared in the scries of "Ostwald's Klassiker."
Page 74. The double integral in the numerator of (IV) is evidently of the
form:
//
(-4)
283
284
ADDENDA.
where the contour of the field of integration (A) is defined by means of the
relations:
a < yiUi < 0, < 2/1 < 1 and < (/o < 1.
The field of integration is thus the area swept out by the hyperbola
y,(/2 = a, the straight line y-z = 1, the hyperbola yiy-i = /8 and the straight
line 2/1 = 1.
Changing the variables by means of the transformation:
2/12/2 = y = <piy, 2) and I - yi = z{l - y) = \p{y, z)
we get the following new double integral
FW^y- z)i '/'(^j z)\\J\dydz {\J\ taken as absolute value),
ss
where / is the Jacobian or functional determinant defined by the formula:
For
1^1 =
J =
dip cJip
dyJz
d4^dxp
dy dz
3<^ 3i^ (?<p d\l/
~ dy dz dz dy
2/2
^ - c(^
/, 2),
- 1 - z(l - 2/) ^^-
2/j = 1 - 2(1 - 2/) = >/'(2/, 2),
1 - z 2/(1 - ?/)
[1 - z(l - 2/)P [1 - 2(1 - yW
z
- (1 - y)
1 -y
1 - 2(1 - y)
The transformation in a double integral implies in general three parts
(1) the expression of ^(2/12/2) in terms of y, z(2) the determination of the new
system of limits (3) substitution of dyidy^. The solution of the third part we
just gave above. The solution of the two first is purely algebraically. The
first part is a straightforward simple problem which should present no difficulty
V
V'
=P
c
lu
li'
Hz
V=a
' — ^-
a ji 1 - Q 1
1 SeeGouus.\T: " Mathematical Analysis" (New York, 1904) pages 266-67.
ADDENDA. 285
whatsoever to the student and which in conjunction with (3) brings the in-
tegrands on the form given in formula (V).
The easiest waj' to determine the new system of Hmits is probabl}- by con-
structing the contour in the new field of integration. The hyperbolas
yoji = a and ijiiji = /3 are in the new field of integration changed into the two
straight lines y = a and y = which determine the limits for the variable y.
A mere inspection of the expressions for <p(y, z) and \p{y, z) shows that the two
straight lines y-i = 1 and (/i = 1 become in the new field 2 = 1 and z =
which are the limits for z.
The contour (Ai) simply becomes a rectangle bounded by the straight
lines z = 0, y=(3, z = l and y = a. The complete transformation finally
brings the numerator on the form as given in (V).
Page 75. The question put by Mr. Bing is simply the determination of a
future event by means of Bayes's Rule. The limits a and /3 become and 1
respectively and the contom- of the field of integration simply becomes the
area bounded by yiy^ = 0, /yo = 1, yiyo = 1 and yi = 1, i. e., the area enclosed
between the two axis, the line 2/2 = 1, the hyperbola 2/1^/2 = 1 and the Una
2/i = 1. The transformed contour becomes a square with side equal to unity.
Chapter VH.
Page 83. The criticism by the English empiricists is to a certain extent
due to a misconception of the Bernoullian Theorem. "This theorem," Venn
says, "is generally expressed somewhat as follows: That in the long run all
events will tend to occur with a frequency proportional to their objective
probabihties." Any one giving careful attention to the deduction of the famous
theorem will, however, readily notice the fallacy of such a view. Not the
actual absolute frequencies of the events but the mathtmalical expectations of
such events are proportional to the a priori mathematical probabihty p. The
fallacy of Mr. Venn hes in his confusing an actual event with its mathematical
expectation. In other words, he makes the Bernoullian Theorem appear as
a regular hypothetical judgment whereas as a matter of fact it is a simple
probability judgment. If one is to take such an erroneous view of the Ber-
noullian Theorem one may even be reconciled with another startling statement
by Venn that "If the chance (against the happening of a certain event) be
1,023 to 1 it undoubtedly will happen once in 1,024 trials."
For a clear presentment of the empirical methods and their relation to
mathematical probabilities and deductive methods see v. Bortkiewicz
"Kritische Betrachtungen zur theoretischen Statistik" (Jahrb. f. X.-Oe. u.
Stat. 3 Folge, Ed. 8, 10, 11) and "Die statistischen Generalisationen " (Sci-
entia, Vol. V). v. Bortkiemcz is but one of the brilliant school of Russian
statisticians who has made a thorough study of the philosophical aspects of
statistics. The induction method of J. S. Mill is carried much farther and
put on a far .sounder basis than that originally given by Mill in the brochure
" Die Statistik als Wissenschaft " by A. A. Tschuproff as well as in his Russian
text " Researches on the Theory of Statistics." The main ideas of the Russian
writers are also found in Kaufmann's "Theorie and Methoden der Statistik"
(Tubingen, 1913).
286 ADDENDA.
Chapter IX.
Page 95. For a closer approximation of /;! see Forsyth, A. R., "On an
Approximate Expression for .r!" (Brit. Ass. Rep., 1883). Page 107. In this
discussion it must be remembered that the variables are independent of each
other. The formula: €{ka) = At (a) is self evident, but may be proved as
follows :
e{ka) = ksp = ke(a), e[ka — e(ka)f = kh'[a — c(a)p = kh^{a), or ({ka) = ke(a).
Page 115. See also a similar discussion by Westergaard in " Mortalitiit und
Morbilitiit" (Jena, 1902), page 187.
Cuapteu XI.
The still unfinished series of monographs by Charlier are found in various
volumes of Meddclande Jrdn Lunds Aslronomiska Obaervatorium (Lund,
Stockholm) and in Svoiska Aktuariefdreningens Tidsskrifl (Stockholm).
Page 137. Since all statistical characteristics to greater or less extent are
effected with mean errors due to sampling it is of importance to be able to
determine such mean errors in simple algtibraic terms. We shall for the present
confine ourselves to the mean and the dispersion. The mean error in the mean,
Mb in a Bernoullian Series is given by the formula:
-N/e2(mi) + e^im2) + ■■■ e^irriTi) ^Nspoqo a
The mean error of the dispersion is somewhat difficult to obtain by elementary
methods since it involves the determination of the mean error of the mean error.
The mean error square of the mean error square ma)' be gotten by a process
similar to that of Laplace in § 65-66 by the introduction of the parameter, i,
in the expression for o? and «■* in c[(a — xp)- — spq^-. After several reductions
this latter exjjression may be brought to the form: 2(spo3o)^ = 2cr^ (approx.).
For the diyi)cr.sion we have:
V2iV
This formula will be proven under the discussion of frequencj^ curves.
Chapter XIII.
Page 184. The Danish engineer, AndriC, discovered a similar correlation
formula about the same time as Bravais.
Chapter XIV.
Page 196. Viewed from the standpoint of elementary errors the expression
for the frequency curve, <^(.r), may be tlerived in the following fashion: —
Let us arrange the elementary errors in small equidistant groups or intervals
of magnitude, a, and assume; ihat all the elementary errors when situated in
the same interval are of (^qual size, an assumption which is always permissible
for small values of o. This means that when r is an integral positive number
all the errors located in the interval ra — 3^a and ra. + \^o. must be of
ADDENDA. 287
equal size and equal to ra. The relative frequency or the probability of such
errors is first of all proportional to the interval, a, and depends in the second
instance upon a certain — so far unknown — function, /(ra), of the quantity
ra. For a particular error source, say Qv, we may therefore express the
probability of the occurrence of an error, ra, as
afvira)
where fv is the unknown function for which we make no other assumptions
than those which follow immediately from the i)roperties of mathematical
probabilities, i. e.
r = <x>
OSf,(ra) S 1 and y> , Mr a) = 1 for v = 1, 2, 3, . . . s.
r = — CO
Consider now for the moment the following expression
7* = OO
FrU) = ^j a fAray"^' where i = V^^
The coefficient of e''°-^^ in the sum is evidently the probability for the occur-
rence of an error ra from the error source Qr.
The probability of the occurrence of an error ra from another error source,
Qu, may similarly be expressed as
Fu(a:) = ^, afuira) e'"^^
and so on for all the s independent error sources, which we assumed to be
operative on our statistical object.
The probability that the resulting sum from the various combinations into
which the elementarj errors from the s sources may enter is found by forming
the product
!■ = S
* (co) = II Fv{^) = /i'lM F^{^) F^io,) . . . F,i^)
v= 1
in accordance with the multiplication theorem of mathematical probabilities
Writing the above products as
* (co) = a [(p (0) + (p (a) e''"' + (p (2a) e^acoi _^ ^ (3^) ^,3aa;i ^
+ <pi-a) e-'^'^^ + (p (-2a) g-^^w^ + (p ( -3a> «--^<^'^' + . . .]
we notice that the coefficient, a(p(ja), of c'"'^"' is the probability that the
288 ADDENDA.
elementary errors from the s error sources will enter into such a combination
that their sum will fall between ra — 3^a and ra + J^a.
.Multii)lyiiifj: on both sides of the above equation by c""'""^'^' (considering
aoi as tile incle[)endent variable) and integrating with respect to aco between
the limits — x and + tc we find that
a(p (ra) = / * (co) e-'"'^"^d (aco).
In the above integral aco is the independent variable. If co is chosen as
the independent variable, we have
a(p {ra) = ~- \ * (co) e"'"'^'^*' f/co.
2x,7 — x/a
If now we let ra = x and let a converge towards zero, in which case
a = clx, we evidently find that
dx r + =°
(P (x) dx = I * (co) e-"-""' do:
2-K J _ oo
is the infinitely .small probability that the sum of the elementary errors
from the s sources will fall in the infinitely small interval x — Hfix and
X + I4dx.
It is evident that by introducing a new function i^ (co), defined by the rela-
tion #(co) = \/2x'A(w), the above equation reduces to (5b) on page 195 if
we let <p(x) =f(x).
Also by writing
— co-r — C0+ — C0+... / —
,, lL ^|2'^^13'^^---= V2x\Kco)
we obtain the general form of the frequency curve as shown on the bottom of
page 196.
Chapter XVI
Page 237. The footnote mentions McAlister as one of the earliest investi-
gators who used the geometric mean as the most probable value of the
observations. I find, however, that McAIister's work was anteceded by that
of Thiele and Gram. Thiele as early as 1866 used the geometric mean as the
most probable value in a series of estimates of the distances of double stars,
in a Danish monograph entitled IJndersogelse af Omldbsbevftgelsen i Dohheltstjer-
nenyslemet Gamma Virfjinis. (Investigation on the movements of rotation in
the double star system Gamma Vi giiiis.)
THE MATHEMATICAL THEORY OF PROBABILITIES
By ARNE FISHER
ERRATA
Page 232, line 2:
For ki = Tik^ read ki = Viki^
Page 237, lines 16 and 17 to read:
/-(-oo /^oo If log X — m \2
x'F{x)dx = {nyl2-K)-''Njx'e ^V » ' dx
— 00 O
/-(_oo _]/£— m\2
— 00
on the assumption that z or log x is normally distributed.
Page 252, Ime 13: +48 to read -48.
ADDENDA. 289
Page 237. The integral in question may be evaluated as follows:—
Let t = {z — m) -.n, ornl -= z — m and ndl = dz.
Hence the integral may be written as
l)(nt + m) -'-
]\J'g(r+ Dm
— 00
+ CO
I- Dm p
/2
ntir + 1) - 2
V2? ' '^ ^ '^^
+ 00
~V^' ' J ' dt
— <x>
If we now let [/ - n{r + 1)] = u, we have dt = du, and the last expression
reduces to
V2^ J e - t/a = A^e e 2 , since the
+
latter integral / ^ 2 ,_
J e " f^M = V2ic
Now in the press Ready for distribution about August 1, 1922.
An Elementary Treatise on
Frequency Curves
And their Application to the Construction
of Mortality Tables
By ARXE FISHER
English translation by E. A. VIGFUSSON
With an Introduction by Professor RAYMOND PEARL
Department of Biometry and Vital Statistics of
the Johns Hopiiins University
(Pp. 225 +XV)
This book falls into two parts of which the first gives an elemen-
tary presentation of the theory of frecjiiency finictions along similar
lines as those developed by ]Mr. Fisher in his book on Probabilities.
The second part, as pointed out in Mr. Vigfusson's preface, con-
stitutes an entirely new departure in the analysis of mortality
statistics. The author has set himself the difficult task to con-
struct a comjilete mortality table from mortuary records by sex,
attained age at death and causes of death, but u-ithout knowledge
of the exposed to risk at various ages. The accomplishment of this
problem has been made possible by means of a biological hypothesis
and a proper classification of the causes of death upon biologiccd
principles. Once accepted the proposed hypothesis will make it
possible to study the laws of human mortality in directions which
hitherto have been regarded as impossible. .Mr. Fisher has applied
his new method to more than 25 population or occupational groups
and gives in this book the detailed results of some of his investiga-
tions in the way of 6 complete mortality tal)les for Michigan
Males (lf)09-191o), Massachusetts Males (1914-191(5), American
Locomotive Engineers (1913-1917), American Coal Miners
(1913-1917), Japanese Assured Males (19U-1917) and White
Industrial Assured Males of the Metropolitan Lite insurance Co.
(1911-1910).
As a systematic treatise on frequency curves and tiieir applica-
tion to mortality studies this book should prove of great practical
^•alue not only to students of statistical methods, hut to actu-
aries, statisticians, health officers, biologists and students of
general science as well.
Comments of Specialists
"Orthodoxy aiul discovery .ire ;is incoinpatihle intellectually as oil and water are
physically, a cosmic law often overlooked by our "safe and sane" scientific gentry.
This book is an outstanding feature that this law is still in operation
It may fairly be regarded as finulameutalhj the most significant advance in actuarial
theory since Ilalley It opens out wonderful possibilities of research
on the laws of inortality in directions which hitherto have been wholly impossible
of attack. The criterion by which the significance of a new technique in any branch
of science is evaluated, is just this at tlie degree to which it opens up new fields of
research. By this criterion Fisher's work stands in a high and secure position."
(Extract from Professor Pearl's Introduction.)
"Fishers novel method has injected new blood in the old body of actuarial
science." {C. liurrau.)
"This new and novel idea meets in reality a very frequent need. It represents
a supplement to the former tools of the actuary and makes possible the utilization
of a statistical material, which according to the requirements of the older systems
was considered as being of no value."
(Extract from Forsikringstidendc's report of discussion in the Norwegian Ac-
tuarial Society, .lune, 1920.)
"Since particularly in industrial statistics, or in general statistical in<juiries under
war conditions, it is easier to obtain accurate data of deaths at ages than of ex[)osed
to risk the success of the method is encouraging The subject is one of
peculiar interest at the present time."
{Journal of the Royal Statistical Socirti/, London, I'JIS.)
From the thcorclical point of view the method of Fi.sher is interesting. His
proposal to decompo.se the mortality according to the dift'erent cau.ses of death is
entirely in conformity with the spirit of modern science which aims to analyze the
j)heii()mena by their difTerent ial jiarts. From the practical jxiint of view the method
is readily applied provided one has a double entry tal)le of the mortuary records by
age and cause of death.
{Bulletin de l' Association drs Actuaires Suisses. The Journal of the Association
of Smss Actuaries, 1010.)
C, ' ■'
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