Let $(Z_n)_{n≥0}$ be a Galton Watson Process with offspring distribution $(p_n)_{n≥0}$
satisfying:
$p_0,p_2>0,$
$p_1∈[0,1)$ and $p_n=0$
otherwise.
Find the extinction probabilty q.
My attempt:
I know $p_0 = P$($0$ offspring), $p_1 = P(1$ offspring), $p_2 = P(2$ offspring).
And therefore generating function has to be $f(s) = p_0 + p_1 s + p_2 s^2$
I know that the extincition probabilty ist the solution of the generating function, so it has to be $q = p_0 + p_1 q + p_2 q^2$ but i stuck at this point. How do i solve this? Is there any other way?
Thanks in advance!
Best Answer
The equation in nothing but $p_2(q-1)(q-\frac {p_0} {p_2})=0$. Hence the extinction probability is $\frac {p_0} {p_2}$. Note: $q=1$ is always a root of the equation $f(s)=s$ where $f$ is the generating function. Knowing that $1$ is a root of a qudarritc equation makes it easy to find the other root.