Probability Generating Function of Zn

Question

Given the following Probability Generating Function:

$G(s)=1-α(1-s)^β$

where $0≤α≤1$ and $0$ < $ β≤1$.

Are we able to derive the Probability Generating Function $G_n(s)$ of the random variable $Z_n$ in the usual notation where $G_n(s)=G(G_{n-1}(s))$ and thus find an expression for $P(Z_n=0), P(Z_n=1)$, and $P(Z_n=2)$?

I tried finding $E[Z]$ by findng $G'(1)$ but ended up with $αβ(1-s)^{β-1}$, where $(1-s)$ will be 0 and $\beta -1 ≤ 0$ making it undefined.

I was however, able to find the probability of ultimate extinction (treating this as branching process) to be a geoemetric series in power of $\alpha$ into:

$G_n(0) = 1-\alpha ^{\frac{1- \beta ^n}{1-\beta}}$

But I'm not sure how that would be helpful in determining the PGF of $Z_n$ and the respective probabilities.

Answer 1

Using induction on $n$, you easily get $G_n(s)=1-\alpha^{(1-\beta^n)/(1-\beta)}(1-s)^{\beta^n}$ (reading $1-\alpha^n(1-s)$ for $\beta=1$). As for the expected values, you've got a correct result; these don't exist (i.e. are $\infty$) if $\beta<1$.