ABSTRACT In this pedagogical article, distributional properties, some surprising, pertaining to the homogeneous Poisson process (HPP), when observed over a possibly random window, are presented. Properties of the gap-time that covered the termination time and the correlations among gap-times of the observed events are obtained. Inference procedures, such as estimation and model validation, based on event occurrence data over the observation window, are also presented. We envision that through the results in this article, a better appreciation of the subtleties involved in the modeling and analysis of recurrent events data will ensue, since the HPP is arguably one of the simplest among recurrent event models. In addition, the use of the theorem of total probability, Bayes’ theorem, the iterated rules of expectation, variance and covariance, and the renewal equation could be illustrative when teaching distribution theory, mathematical statistics, and stochastic processes at both the undergraduate and graduate levels. This article is targeted toward both instructors and students.

中文翻译：

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停留在齐次泊松过程中


摘要 在这篇教学文章中，介绍了在可能随机的窗口中观察时与齐次泊松过程 (HPP) 相关的分布特性，有些令人惊讶。获得覆盖终止时间的间隙时间的属性以及观察到的事件的间隙时间之间的相关性。还介绍了基于观察窗口上的事件发生数据的推理过程，例如估计和模型验证。我们预计，通过本文的结果，将更好地理解周期性事件数据建模和分析中涉及的微妙之处，因为 HPP 可以说是周期性事件模型中最简单的模型之一。此外，在本科生和研究生教授分布理论、数理统计和随机过程时，总概率定理、贝叶斯定理、期望、方差和协方差的迭代规则以及更新方程的使用可以起到说明作用。水平。本文面向教师和学生。