Consider the following modification of a branching process: A mature individual produces children according to the generating function g(t). However, an individual becomes mature with probability α and dies before maturity with probability 1 - α. Throughout X(0) = 1, that is, we start with one immature individual.
(a) Find the generating function of the number of individuals in the first two generations.
(b) Suppose that the offspring distribution is geometric with parameter p. Determine the extinction probability.