Let X1, X2, . . . , Xn denote the first n interarrival
Chapter , Problem 40(choose chapter or problem)
Let X1, X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn = ni =1 Xi . (a) What is the interpretation of Sn? (b) Argue that the two events {Sn t} and {N(t) n} are identical. (c) Use part (b) to show that P{Sn t} = 1 _n1 j=0 et(t) j/j! (d) By differentiating the distribution function of Sn given in part (c), conclude that Sn is a gamma random variable with parameters n and . (This result also follows from Corollary 5.7.2.) *
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