Abstract

Recently, the degenerate Poisson random variable with parameter \(\alpha > 0\), whose probability mass function is given by \(P_{\lambda}(i) = e_{\lambda}^{-1} (\alpha) \frac{\alpha^{i}}{i!} (1)_{i,\lambda}\) \((i = 0,1,2,\dots)\), was studied. In probability theory, the zero-truncated Poisson distributions are certain discrete probability distributions whose supports are the set of positive integers. These distributions are also known as the conditional Poisson distributions or the positive Poisson distributions. In this paper, we introduce the degenerate zero-truncated Poisson random variables whose probability mass functions are a natural extension of the zero-truncated Poisson distributions, and investigate various properties of those random variables.

中文翻译：

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简并零截断泊松随机变量

摘要

最近，具有参数\（\ alpha> 0 \）的退化泊松随机变量，其概率质量函数由\（P _ {\ lambda}（i）= e _ {\ lambda} ^ {-1}（\ alpha）给出研究了\ frac {\ alpha ^ {i}} {i！}（1）_ {i，\ lambda} \） \（（i = 0,1,2，\ dots \\）。在概率论中，零截断的泊松分布是某些离散的概率分布，其支持是一组正整数。这些分布也称为条件泊松分布或正泊松分布。在本文中，我们介绍了退化的零截断的Poisson随机变量，其概率质量函数是零截断的Poisson分布的自然扩展，并研究了这些随机变量的各种性质。