## A Couple of Questions Re: Birth and Death Processes

Hello,

I have a few homework questions regarding birth & death processes (CTMCs) that I'm having trouble solving:

1) Suppose a CTMC has a death rate of 0, and birth rate of n*lambda (proportional to the current population) - i.e. Yule Model. At a deterministic time T, the process stops growing and is replaced by an emigration process in which departures occur according to a Poisson process with intensity γ. If Y denotes the time taken after T for the population to vanish, find the probability density function of Y.

For this question I was considering using total probability formula, P(T>t|X(T)=n)*P(X(T)=n) summed over all n. However, I can't seem to figure out a formula to solve for the probability that the Birth & Death process has a specific population at a specific time.

2) Consider a birth-death process with birth rates λn = nλ and death rates
µn = nµ. Show that E(X(t)|X(0) = 1) = e^(λ−µ)t.

I have no idea how to approach this one. It seems to require solving for matrix P(t) = e^Rt, but I don't know how one would solve this given R is an infinity by infinity matrix.

Any help would be much appreciated. Thanks!