Probability MT22, Branching processes

@Define the “offspring distribution” of a branching process.

What assumption is made about individuals reproducing in a branching process, other than having the same offspring distribution?

If the offspring distribution has probability generating function $G(s)$, what is the probability generating function for the number of individuals in the $n$-th generation?

What’s the basic gist of proving the fact $G _ {n+1}(s) = G(G _ n(s))$ for a branching process?

If each individual in a branching process has a mean of $\mu$ children, what is the expected number of individuals in the $n$-th generation?

What’s the probability that a branching process with generating function $G(s)$ dies out?

@Prove that if a branching process has offspring distribution given by probability generating function $G$ then the probability generating function for the number of individuals at generation $n$ (∆generation-n-generating-function) is given by

Let $X _ n$ be the number of children in the $n$-th generation of a branching process, and suppose the mean number of children of a single individual is $\mu$. @Prove that (cf. ∆expected-generation-size)