(a) Consider a Galton-Watson process . Prove that the extinction probability is the smallest non-negative solution of the equation where . [You should prove any properties of Galton-Watson processes that you use.]
In the case of a Galton-Watson process with
find the mean population size and compute the extinction probability.
(b) For each , let be a random variable with distribution . Show that
in distribution, where is a standard normal random variable.