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lec0224 - Poisson Process Special Case of Many Things It is...

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Poisson Process: Special Case of Many Things It is useful to be aware that a Poisson process is a special case of several important stochastic processes. That leads to different equivalent definitions of a Poisson process, as in Definitions 5.2 and 5.3 of the Ross text. It also leads to different ways to analyze a Poisson process. 1. A Point Process and a Counting Process A point process on the positive half line, i.e., on the interval [0 , ), is a random distribu- tion of points on the positive half line. We may specify the distribution in three ways: (i) by specifying the distribution of the locations of the points, (ii) by specifying the distribution of the intervals between successive points and (iii) by specifying the distribution of the associated counting process. Let S n be the location of the n th point, where S 0 0 (without there being a 0 th point). Let X n S n - S n - 1 be the interval between the ( n - 1) st point and the n th point. Let the associated counting process be defined by N ( t ) max { k 0 : S k t } , t 0 . In other words, a point process may be specified in three ways, via the stochastic processes: (i) { S n : n 0 } , (ii) { X n : n 1 } and (iii) { N ( t ) : t 0 } . The first representation { S n : n 0 } is the typical form for a point process . The last representation { N ( t ) : t 0 } is the typical form for a counting process . A picture makes this clear; see Figure 1. Figure 1: A Sample Path of a Counting Process.
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