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Chapter 9 Poisson processes - Yale University

Page 1 Chapter 9 Poisson processesThe Binomial distribution and the geometric distribution describe the behavior of tworandom variables derived from the random mechanism that I have called coin tossing . Thenamecoin tossingdescribes the whole mechanism; the namesBinomialandgeometricreferto particular aspects of that mechanism. If we increase the tossing rate to m tosses per sec-ond and decrease the probability of heads to a small p, while keeping the expected numberof heads per second fixed at Dmp, the number of heads in atsecond interval will haveapproximately a ;p/distribution, which is close to the Poisson . t/. Also, the num-bers of heads tossed during disjoint time intervals will still be independent random the limit, asm!1, we get an idealization called aPoisson process. Poisson process< > Poisson process with rate on[0;1/is a random mechanism that gener-ates points strung out along[0;1/in such a way that(i) the number of points landing in any subinterval of lengthtis a random variable witha Poisson .]]

Chapter 9 Poisson processes Page 4 Compare with the gamma.1=2/density, y1¡1=2e¡y 0.1=2/ for y >0: The distribution of Z2=2 is gamma (1/2), as asserted. Note: From the fact that the density must integrate to 1, we get a bonus:

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