Some results on r-truncated degenerate Poisson Random Variables

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

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 .

r-TRUNCATED DEGENERATE POISSON RANDOM VARIABLES
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

Now, we observe that
.
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.
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! .

CONCLUSION
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.