Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function:
$$p(i)=pq^i\qquad i\geq0$$
where $q=p-1$. Find the probability generating function. Find the mean.
Would really appreciate if someone could direct me on how to find the mean, but the focus is how to find the pgf.
I know that $G_n(s)=G_{n-1}(G(s))=G(G_n(s))$. In other words, I need to find some $G(s)$ and then plug this into itself $n$-times, or guess some pattern. I have made very little headway.
$$G(s)=\sum^{\infty}_{i=0} p(i)s^i=\sum p (qs)^i$$
I assume that $|qs|<1$; therefore,
$$\sum p (qs)^i=\frac{p}{1-qs}.$$
However, upon plugging this into itself, I fail to notice anything meaningful. What am I missing, or what did I do wrong? How should I proceed?
Denote the mean population for generation $n$ by $m_n$. Then $$m_1 = \mathbb E[X_1] = \lim_{s\uparrow 1}G'(s) = lim_{s\uparrow1}\frac{p(1-p)}{(1-(1-p)s)^2} = \frac{1-p}p.$$ Since $G_n(s) = G_{n-1}(G(s))$ (where $G_n(s)$ denotes $G$ composed $n$ times), we have $$ G_n'(s) = G_{n-1}'(G(s))G'(s).$$ Hence $$\begin{align*} m_n &= \lim_{s\uparrow 1} G_n'(s)\\ &= \lim_{s\uparrow1} G_{n-1}'(G(s))G'(s)\\ &= m_{n-1}m_1,\end{align*}$$ and from this recursion it is apparent that $$ m_n = m^n = \left(\frac{1-p}p\right)^n.$$
In general determining the distribution of $X_n$ is very difficult. I don't know how to compute $G_n(s)$ either.