Consider a branching process with offspring distribution Geometric($\alpha$); that is, $p_{k} =α(1−α)^k$ for $k≥0$.
a) For what values of $α ∈ (0, 1)$ is the extinction probability $q = 1$.
b) Use the following proposition to give a formula for the extinction probability of the branching process for any value of the parameter $α ∈ (0, 1)$.
My work :
for part a, I calculated the prob. generating function and from there calculate the extinction prob. and equate it to $q$. The equation I got was: $$ q = \frac{\alpha}{1-(1-\alpha)q}$$
I replaced q with 1 and I was going nowhere from here.
So, I tried Proposition 1 which gave me
$$\mu = \frac{1-\alpha}{\alpha}\leq1$$
$$\alpha \geq \frac{1}{2}$$
How to approach part b?
The extinction probability is the smaller of the two roots in $[0,1]$ of the equation $ \sum\limits_{k=0}^{\infty}p_kz^{k}=s$. In this equation this equation becomes $\frac {\alpha} {1-(1-\alpha)s} =s$ or $(1-\alpha)s^{2}-s+\alpha =0$. You can wriet thsi as $(1-s) (\alpha -(1-\alpha) s)=0$ so the extinction probability is $\frac {\alpha} {1-\alpha}$.