# Question about a stochastic process

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 $0$.

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