A study on properties of degenerate and zero-truncated degenerate Poisson random variables

TL;DR

This paper explores the properties of degenerate and zero-truncated degenerate Poisson random variables, extending classical Poisson concepts using degenerate versions and analyzing their moments, distributions, and generating functions.

Contribution

It introduces and investigates the properties of zero-truncated degenerate Poisson random variables, extending existing degenerate Poisson models with new distributional and moment results.

Findings

Derived the expectation and variance of zero-truncated degenerate Poisson variables.

Established the probability-generating function as the generating function of degenerate Lah-Bell polynomials.

Analyzed the distribution of sums of independent degenerate zero-truncated Poisson variables.

Abstract

Carlitz [2] initiated a study on degenerate versions of Bernoulli and Euler numbers which has been extended recently to the researches on various degenerate versions of quite a few special numbers and polynomials. They have been explored by using several different tools including generating functions, combinatorial methods, $p$-adic analysis, umbral calculus, special functions, differential equations and probability theory as well. \par The degenerate Poisson random variables are degenerate versions of the Poisson random variables. In [6], studied are the degenerate binomial and degenerate Poisson random variables in relation to the degenerate Lah-Bell polynomials. Among other things, it is shown that the rising factorial moments of the degenerate Poisson random variable are expressed by the degenerate Lah-Bell polynomials. Also, it is shown that the probability-generating function of the degenerate Poisson random variable is equal to the generating function of the degenerate Lah-Bell polynomials. The zero-truncated Poisson distributions (also called the conditional or the positive Poisson distributions) are certain discrete probability distributions whose supports are the set of positive integers. In [10], the zero-truncated degenerate Poisson random variables, whose probability mass functions are a natural extension of the zero-truncated Poisson random variables, are introduced and various properties of those random variables are investigated. Specifically, for those distributions, studied are its expectation, its variance, its n-th moment, its cumulative distribution function and certain expressions for the probability function of a finite sum of independent degenerate zero-truncated Poisson random variables with equal and unequal parameters.