In a Galton-Walton branching process, the offspring distribution is given by

$\displaystyle \mathbb{P}(X = r) = p_0 \ if \ r=0, (1-p_0)pq^{r-1}\if\ r=1,2,...$

where $\displaystyle 0 < p < 1, q=1-p, and \ 0 < p_0 < 1$

Find the probability of extinction (assuming $\displaystyle N_0=1$)

i did:

find the generating function first G(s).

$\displaystyle G(s)= \sum_{i=0}^{\infty} s^k.p_k $

$\displaystyle = p_0 + (1-p_0)ps\frac{1}{1-qs}$

now to find e so i need to find smallest non negative solution of s=G(s)

$\displaystyle s=p_0 + (1-p_0)ps\frac{1}{1-qs}$

and i get:

$\displaystyle s=\frac{1}{4}\frac{q^2+1-p_0q-p_0^2}{(1-p)^2}$

however the next part of the question asks you to find, supposing $\displaystyle N_0$ has a poisson distribution with mean $\displaystyle \lambda$. Supposing also that $\displaystyle p_0 < q$ the probability of ultimate extinction is

$\displaystyle exp({-\lambda(1-p_0/q)})$

I know that if $\displaystyle N_0=m$ then it will be $\displaystyle s^m$

however i cannot see how to get another distribution into the answer