52. Let Xi, i = 1, 2,..., be the interarrival times of the renewal process {N(t)}, and let Y , independent of the Xi, be exponential with rate λ.

(a) Use the lack of memory property of the exponential to argue that P{X1 +···+ Xn < Y} = (P{X

n

(b) Use part

(a) to show that E[N(Y)] =

E[e−λX]

1 − E[e−λX]

where X has the interarrival distribution.