So I'm trying to construct a probability generating function for the following scenario:
1/5 of a rabbit population does not reproduce. 4/5 have 3 offspring each, and the probability of male or female is equal. How do I construct a probability generating function for this branching process?
I know the format of the probability generating function, $\phi(s)$ can be represented as:
$$\sum_{k=0}^\infty {s^k}{P_k}\qquad\qquad 0\le s \le 1$$
Where $k$ is the number of new offspring, $P_k$ is the probability that each member of the population produces $k$ offpsring. Assume the rabbits are monogamous.
I don't know how to set up $\phi(s)$ taking into account the random generation of males and females. $\phi(s)$ represents the male population per generation.
Let $Z_{n,j}$ be i.i.d. with common distribution $$ \mathbb P(Z_{n,j}=0)=\frac15 = 1- \mathbb P(Z_{n,j}=3)$$ and $X_{n,j}$ be i.i.d. with common $\mathrm{Ber}\left(\frac12\right)$ distribution for positive integers $n$, $j$, and $k=1,2,3$. Assume that we start with one male and one female rabbit. The population in generation $n$ is given by $Z_0=1$ and $$Z_n=\min\left\{\sum_{j=1}^{Z_{n-1}} Z_{n,j}\sum_{k=1}^3X_{j,k},\quad \sum_{j=1}^{Z_{n-1}} Z_{n,j}\sum_{k=1}^3(1-X_{j,k}) \right\}$$ for $n\geqslant 1$. That is, $Z_n$ is the number of reproducing pairs generated as offspring from generation $n-1$, times the number of offspring generated by each pair.