Let {Zn}n∈N0 be a simple branching process which starts from one individual. Each individual has exactly three children, each of whom survives until reproductive age with probability 0 < p < 1, and dies before he/she is able to reproduce with probability q = 1 - p, independently of his/her siblings. The children that reach reproductive age reproduce according to the same rule.
1. Write down the generating function for the offspring distribution.
2. For what values of p will the population go extinct with probability 1.
(Hint: You don't need to compute much. Just find the derivative P0(1) and remember the picture from class →)1.