Let $X_n$ denote the population size in a branching process $\{X_n, n\geq 0\}$. Assume that $X_1=Y_1$ has the distribution $P(Y_1=0)=P(Y_1=1)=1/2$. Answer the following:
a. Find the probability generating function of $Y_1$.
b. Show that the probability generating function of $X_n$ is $G_n(s)=1-1/2^n +s/2^n$, $s\in\mathbb{R}$.
c. Compute the extinction probability of the population starting from one individual at time $0$.
