退化和零截尾退化Poisson随机变量的性质研究
A study on properties of degenerate and zero-truncated degenerate Poisson random variables
摘要：
Carlitz [2]开创了Bernoulli数和Euler数的退化形式的研究，近年来又扩展到许多特殊数和多项式的各种退化形式的研究。他们已经探索了使用几种不同的工具，包括生成函数，组合方法，$p$ -adic分析，本底演算，特殊功能，微分方程和概率论以及。退化泊松随机变量是泊松随机变量的退化版本。文[6]研究了退化二项和退化Poisson随机变量与退化Lah-Bell多项式的关系。除其他事项外，它表明，退化泊松随机变量的上升阶乘的时刻表示的退化Lah-Bell多项式。并证明了退化Poisson随机变量的概率母函数等于退化Lah-Bell多项式的母函数。零截尾泊松分布（也称为条件泊松分布或正泊松分布）是一类支持集为正整数的离散概率分布。在文献[10]中，引入了零截尾退化Poisson随机变量，其概率质量函数是零截尾Poisson随机变量的自然推广，并研究了这类随机变量的各种性质。具体地说，研究了这些分布的期望、方差、n阶矩、累积分布函数以及具有相等和不等参数的独立退化零截尾Poisson随机变量的有限和的概率函数的表达式。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
