Consider a conditional Poisson process where the distribution of A is the gamma distribution with parameters m and a that is, the density is given by - g(A) = aea (a)/(m 1)', 0 < x < .

(a) Show that m+n-1 P{N(t) = n}= n (m + n 1 ) (+) (+) * m n0. a+

(b) Show that the conditional distribution of A given N(t) = n is again gamma with parameters m + n, a + 1.

(c) What is - lim P{N(th) N(t) = 1 | N(t) = n}/h? h-0