1. Consider a branching process with reproduction mean m 1. Suppose also, as before, that X(0) = 1.
(a) What is the probability of extinction?
(b) Determine the expected value of the total progeny.
(c) Now suppose that X(0) = k, where k is an integer ≥ 2. What are the answers to the questions in (a) and (b) now?
2. The following model can be used to describe the number of women (mothers and daughters) in a given area. The number of mothers is a random variable X ∈ Po(λ). Independently of the others, every mother gives birth to a Po(µ)-distributed number of daughters. Let Y be the total number of daughters and hence Z = X + Y be the total number of women in the area.
(a) Find the generating function of Z.
(b) Compute E Z and Var Z.