Let {X(n), n ≥ 0} be the usual Galton-Watson process, starting with X(0) = 1. Suppose, in addition, that immigration is allowed in the sense that in addition to the children born in generation n there are Zn indi- viduals immigrating, where {Zn, n ≥ 1}are i.i.d. random variables with the same distribution as X(1).
(a) What is the expected number of individuals in generation 1?
(b) Find the generating function of the number of individuals in genera- tions 1 and 2, respectively.
(c) Determine/express the probability that the population is extinct after two generations.
Remark. It may be helpful to let p0 denote the probability that an in- dividual does not have any children (which, in particular, means that P (X(1) = 0) = p0).