40 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 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 ) n 1 P S, t) 1(t)j j 0

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Question: 40. Let X1, X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn = ni =1

40. 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

n-1 P{S, t)=1(t)j! j=0

d) By differentiating the distribution function of Sn given in part (c), conclude that Sn is a gamma random variable with parameters n and λ. (This result also follows from Corollary 5.7.2.)

n-1 P{S, t)=1(t)j! j=0

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