For the Yule process.

(a) verify that satisfies the forward and backward equations

(b) Suppose that X(0) = 1 and that at time 7 the process stops and is replaced by an emigration process in which departures occur in a Poisson process of rate . Let 7 denote the time taken after T for the population to vanish. Find the density function of 7 and show that E[T] = e.