Chapter 5 - Poisson Processes

Abstract

This chapter focuses on the Poisson and the Poisson process distribution. Poisson distribution is amenable to extensive and elaborate analysis and the behavior of the Poisson is pervasive in natural phenomena as to make the Poisson process a cornerstone of stochastic modeling. The chapter discusses the theorems and basic concepts of the Poisson distribution. The Poisson process entails notions of both independence and the Poisson distribution and can be further categorized as non homogeneous processes and homogeneous processes. The law of rare events explains the common occurrence of the Poisson distribution in nature. The chapter also discusses distributions associated with the Poisson process, the uniform distribution and Poisson processes, spatial Poisson processes, and compound and marked Poisson processes. A model is presented to describe the failure time of a sheet or volume of material subjected to a constant stress σ. The failure time is viewed in two parts, crack initiation and crack propagation.