consider the branching process $\{X_n,n\geq 0\},X_0=1, > P(Y=k)=\frac{1}{2^{k+1}},k = 0,1,2, ...$
show the pgf of Y is $G(s) = \frac{1}{2-s}$
what is the probability of eventual extinction?
find the pgf for $X_n$.
I've shown the pgf for Y. but when solving $G(s)=s$ the two roots are 0 and 2. if $\pi_0$ is the minimal roots then extinction probality would be $0$, but when i calculated $E(Y) = G'(s=1) = 1$ which indicates extinction. Why is it contradictory? Also I do i find the pgf of $X_n$?
$\frac 1 {2-s}=s$ gives $(2-s)s=1$ or $s^{2}-2s+1=0$. This gives $(s-1)^{2}=0$ so $1$ is the only root.