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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.

Chapter 9 Poisson processes Page 3 D.k ¡1/.k ¡2/.k ¡3/:::.2/.1/0.1/ D.k ¡1/! because 0.1/D R1 0 e ¡xdx D1.Compare with the fact that the gamma.k/density in <9.3> integrates to one. ⁄ Exponential distribution Specializing the gamma.k/to the case k D1 we get the density e¡t for t >0; which is called the (standard) exponential distribution.

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