key: cord-0184278-yip70nmz
authors: Kim, Taekyun; Kim, Dae san; Lee, Si-Hyeon; Park, Seong-Ho; jang, Lee-Chae
title: Some results on r-truncated degenerate Poisson Random Variables
date: 2021-04-28
journal: nan
DOI: nan
sha: 5c674b68e0fd2603ba70a01beb672d08c459bb78
doc_id: 184278
cord_uid: yip70nmz

The zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers, which are also known as the conditional Poisson distributions or the positive Poisson distributions. In this paper, we introduce the r-truncated degenerate Poisson random variable with parameter a and investigate various properties of this random variable

It is well known that a random variable X , taking on one of the values 0, 1, 2, . . . , is said to be the Poisson random variable with parameter α > 0, if the probability mass function of X is given by p(i) = P{X = i} = e −α α i i! , (i = 0, 1, 2, . . . ), (see [11, 13] ).

We note that a Poisson random variable indicates how many events occured in a given period of time. Further, a random variable X r , taking on one of the values r + 1, r + 2, . . . , is called the rtruncated Poisson random variable with parameter α > 0, if the probability mass function of X r is given by

where i = r + 1, r + 2, r + 3, . . . , for r ≥ 0. In the special case of r = 0, we obtain the zero-truncated Poisson distribution. On the other hand, the degenerate Poisson random variables with parameter α > 0, whose probability mass function is given by (see (1) )

were studied by Kim-Kim-Jang-Kim in [10] . In addition, the zero-truncated degenerate Poisson random variables with parameter α > 0, whose probability mass function is given by

were studied in [9] . The aim of this paper is to generalize the results on the zero-truncated degenerate Poisson distribution in [9] to the case of the r-truncated degenerate Poisson distribution as a natural extension of the r-truncated Poission distribution. We will obtain, among other things, its expectation, its variance, its n-th moment, and its cumulative distribution function.

One motivation for this research is its potential applications to the cornavirus pandemic. It has been spreading unpredictably around the world, terrorizing many people. Although several vaccines for the coronavirus have been developed and many people are getting vaccinated, they have many unexpected side effects as well. We would like to predict stability of the coronavirus vaccines after r-days the vaccines were shot. For this purpose, we study the r-truncated degenerate Poisson random variables (see (6) ), which has the 'degenerate factor' λ reflecting abnormal situations. Indeed, we think that Theorem 5 is useful in predicting the probability of the coronavirus vaccines becoming stable after the r-days of getting vaccinated. Another motivation is applications of various probabilistic methods in studying special numbers and polynomials arising from combinatorics and number theory. For this, we let the reader refer to [5] [6] [7] [8] [9] [10] 14 and the references therein]. For the rest of this section, we recall the necessary facts that are needed throughout this paper.

For any λ ∈ R, the degenerate exponential functions are defined by

In particular, for x = 1, we denote e 1 λ (t) by e λ (t). In addition, it is convenient to introduce e λ ,r (t), for any integer r ≥ 0, given by

The Bell polynomials are given by

Bel n (x) t n n! , (see [4, 12] ).

In light of (3), the degenerate Bell polynomials are defined in [9] by

Bel n,λ (x) t n n! .

In particular, for x = 1, Bel n,λ = Bel n,λ (1), (n ≥ 0), are called the degenerate Bell numbers. Note that lim

Let X be a Poisson random variable with parameter α > 0. Then the moment of X is given by [11, 13] ).

Let Y be a discrete random variable taking values in the nonnegative integers. Then the probability generating function of Y is given by

where p(i) = P{Y = i} is the probability mass function of Y .

The degenerate Stirling numbers of the second kind S 2,λ (n, k) are defined by (x) n,λ = n ∑ l=0 S 2,λ (n.l)(x) l , (n ≥ 0), (see [5] ), or equivalently by

Here

. Now, we consider the r-truncated degenerate Stirling numbers of the second kind S

[r]

2,λ (n, kr + k) given by

Note that

2,λ (n, k) = S 2,λ (n, k), (see [5] ).

The random variable X λ ,r is called the r-truncated degenerate Poisson random variable with parameter α > 0, if the probability mass function of X λ ,r is given by

Here we must observe that

In addition, we note that

is the probability mass function of the r-truncated Poisson random variable with parameter α > 0.

Let us assume that X λ ,r is the r-truncated degenerate Poisson random variable with parameter α > 0. Then we note that the expectation of X λ ,r is given by

From (7), we obtain the following theorem.

Theorem 1. Let X λ ,r be the r-truncated degenerate Poisson random with parameter α > 0. Then, for λ = − 1 α , the expectation of X λ ,r is given by

.

From (8), we note that

α , we note that the variance of X λ ,r is given by

Therefore, by (10), we obtain the following theorem.

Theorem 2. Let X λ ,r be the r-truncated degenerate Poisson random variable with parameter α > 0.

(1) r+1,λ e λ (α) − e λ ,r (α)

Let us consider the generating function of the moments of the r-truncated degenerate Poisson random variable with parameter α > 0. Then, from (4), we have Thus, by (11) , we get the next theorem.

Theorem 3. Let X λ ,r be the r-truncated degenerate Poisson random variable with parameter α > 0. For n ≥ 0, we have

For x ≥ r + 1, we note that the cumulative distribution function is given by

From (12) and (2), we can derive the following equation.

Theorem 4. Assume that X λ ,r is the r-truncated degenerate Poisson random variable with parameter α > 0. For x ≥ r + 1, the cumulative distribution function of X λ ,r is given by

Let us assume that X (1) λ ,r , X (2) λ ,r , . . . , X (k) λ ,r are identically independent r-truncated degenerate Poisson random variables with parameter α > 0, and let

From the probability generating function of random variable, we note that

By (14), we get

2,λ (n, kr + k) α n t n n! .

On the other hand, (1) λ ,r +X (2) λ ,r +···+X (k)

Therefore, by (15) and (16), we obtain the following theorem.

Theorem 5. Let X (1) λ ,r , X (2) λ ,r , . . . , X (k) λ ,r be identically independent r-truncated degenerate Poisson random variables with parameter α > 0, and let X λ ,r = k ∑ i=1 X (i) λ ,r . Then the probability for X λ ,r is given by

2,λ (n, kr + k), if n ≥ kr + r, 0, otherwise.

We generalized the results on the zero-truncated degenerate Poisson distribution to the case of the r-truncated degenerate Poisson distribution, with its potential applications to the cornavirus pandemic and applications of probabilistic methods to the study of some special numbers and polynomials in mind.

Let X λ ,r be the r-truncated degenerate Poisson random variable with parameter α. Then, for the random variable X λ ,r , we derived its expectation, its variance, its n-th moment, and its cumulative distribution function. In addition, we obtained two different expressions for the probability generating function of a finite sum of independent r-truncated degenerate Poisson random variables with equal parameters.

As one of our future projects, we would like to continue this line of research, namely to explore applications of various methods of probability theory to science, engineering and social science, and to the study of some special polynomials and numbers.

Bernoulli and Eulerian numbers

A degenerate Staudt-Clausen theorem

Some results on tests for Poisson processes

Combinatorial sums involving Stirling, Fubini, Bernoulli numbers and approximate values of Catalan numbers

A Note on a New Type of Degenerate Bernoulli Numbers, Russ

Two variable degenerate Bell polynomials associated with Poisson degenerate central moments

A note on truncated degenerate exponential polynomials

Note on extended Lah-Bell polynomials and degenerate extended Lah-Bell polynomials

Degenerate Zero-Truncated Poisson Random Variables, Russ

A note on discrete degenerate random variables

A. Probability and random processes for Electronical Engineering, 3'

The umbral calculus

Introduction to probability models

New construction of type 2 degenerate central Fubini polynomials with their certain properties

SEOUL 139-701, REPUBLIC OF KOREA Email address: tkkim@kw.ac.kr DEPARTMENT OF MATHEMATICS, SOGANG UNIVERSITY, SEOUL 121-742, REPUBLIC OF KOREA Email address: dskim@sogang.ac.kr* DEPARTMENT OF MATHEMATICS

Acknowledgements: Not applicable.Funding: Not applicable.Availability of data and material: Not applicable.

Competing interests: The authors declare no conflict of interest.