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Probability and Stochastic Processes by Ionut Florescu

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Chapter 10The Poisson Process

Introduction

We now start the study of a stochastic process which is absolutely vital for applications. Random number generation (Lu et al., 1996), queuing theory (Gross and Harris, 1998), quantum physics (Jona-Lasinio, 1985), electronics (Kingman, 1993), and biology (Blæsild and Granfeldt, 2002) are only some of the many areas of application for this process (Good, 1986).

The process is named in honor of Siméon-Denis Poisson (1781–1840), and, as we shall see, it is a continuous-time process (the index set c10-math-0001) with discrete state space c10-math-0002.

We start with basic definitions.

10.1 Definitions

As always, we work on a probability space c10-math-0003 endowed with a filtration c10-math-0004. However, as we shall see, the Poisson process has a discrete state space. Therefore, as mentioned in the probability part of the book, the sigma algebra and the filtration are not essential, and we shall take c10-math-0005 formed from the pre-images of all subsets of . This will allow us to have every set measurable, ...

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