Consider a conditional Poisson process (see Section 2.6 of Chapter 2). That is, let A be a nonnegative random variable having distribution G and let {N(t), 0} be a counting process that, given that A = A, is a Poisson process with rate A. Let G denote the conditional distribution of A given that N(t) = n.

(a) Derive an expression for Gin

(b) Does G, increase stochastically in n and decrease stochastically in /?

(c) Let Y denote the time from until the next event, given that N(t) =n Does Y increase stochastically in and decrease stochas- tically in n?