October 2014
Intermediate to advanced
576 pages
14h 32m
English
Content preview from Probability and Stochastic Processes
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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 
) with discrete state space 
.
We start with basic definitions.
10.1 Definitions
As always, we work on a probability space 
endowed with a filtration 
. 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 
formed from the pre-images of all subsets of . This will allow us to have every set measurable, ...
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