Let Sn denote the time of the nth event of the renewal process {N(t), t 0}

having interarrival distribution F.

(a) What is P(N(t) = n|Sn = y)?

(b) Starting with P(N(t) = n) =

 ∞

0 P(N(t) = n|Sn = y) fSn (y) dy and using that the sum of n independent exponentials with rate λ has the Gamma

(n, λ) distribution, derive P(N(t) = n) when F(y) = 1 − e−λy .