Problem S5.1. Consider a linear birth-death process which includes the possibility of a change in population due to immigration in addition to the change that occurs due to the birth and death of individuals in the population. Assume that at time t the population has n members. Let aAt be the probability that one individual enters the society due to immigration in time t + At. Let BnAt(ynAt) be the probability of a birth (death) in time t+At. Assume that at time t = 0 the population has nm members.

(a) Write the master equation and compute the generating function, F(z,t) = P(n,t), for this process.

(b) Compute the first moment (n(t)), and the variance, (n2(t))-(n(t)), for this process.