A population evolves in generations. Let be the number of members in the th generation, with . Each member of the th generation gives birth to a family, possibly empty, of members of the th generation; the size of this family is a random variable and we assume that the family sizes of all individuals form a collection of independent identically distributed random variables each with generating function .
Let be the generating function of . State and prove a formula for in terms of . Determine the mean of in terms of the mean of .
Suppose that has a Poisson distribution with mean . Find an expression for in terms of , where is the probability that the population becomes extinct by the th generation.