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What is the extinction process of a branching process with $Bin(2, 1-p)$ offspring distribution?

My attempt:

$G_n(s) = E(s^{Z_n})$ is the PGF of $Z_n$, the population size at time $n$

Let $G(s) = E(s^Y)$, $Y~Bin(2, 1-p)$, so the PGF is $G(s) = (ps+q)^n$

Want to solve $G(s)=s$

$G(s) = ((1-p)s+p)^2 = s^2-2s^2p+2sp+s^2p^2s-2sp^2+p^2$

Now I'm having trouble solving:

$s^2-2s^2p+2sp+s^2p^2s-2sp^2+p^2 = s$

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    $$G(s)-s=((1-p)^2s^2+2p(1-p)s+p^2)-s=(1-p)^2s^2+(2p(1-p)-1)s+p^2$$ Surely you can solve $$as^2+bs+c=0$$ especially when $$a+b+c=0\ ?$$ – 2017-02-09

0 Answers 0