Chapter 5 - Poisson Processes

Abstract

This chapter reviews Poisson processes. To begin with, Poisson behavior is so pervasive in natural phenomena and the Poisson distribution is amenable to extensive and elaborate analysis to an extent so as to make the Poisson process a cornerstone of stochastic modeling. Two fundamental properties of the Poisson distribution concern the sum of independent Poisson random variables and certain random decompositions of Poisson phenomena. Consequently, the Poisson process entails notions of both independence and the Poisson distribution. The common occurrence of the Poisson distribution in nature is explained by the law of rare events. Poisson point processes often arise in a form where the time parameter is replaced by a suitable spatial parameter. Also, Poisson events occurring in space can best be modeled as a point process. Conditioned on a fixed total number of events in an interval, the locations of those events are uniformly distributed in a certain way; this provides an important tool for computing certain functional on a Poisson process. Both compound Poisson and marked Poisson processes appear often as models of physical phenomena.