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**RoyalFlush** Let $\displaystyle X_0, X_1, ...$ be a branching process with $\displaystyle \mathbb{P} (X_0 = 1) = 1$, and offspring distribution $\displaystyle \mathbb{P} (Y= k ) = qp^k \ \mbox{for k} = 0,1,2,...$ where $\displaystyle q = 1- p \ \mbox{and} \ p \in (0,1)$.

(1) Prove that the pgf $\displaystyle G_n \ \mbox{of} \ X_n $ is

$\displaystyle G_n (s) = \begin{cases} \frac{n-(n-1)s}{n+1-ns} & p = q \\ \frac{q(p^n-q^n-ps(p^{n-1} - q^{n-1}))}{p^{n+1} - q^{n+1} - ps(p^n - q^n)} & p \neq p \end{cases}$

and

(2) Determine (explicitly) the pdf $\displaystyle H_n $ of $\displaystyle X_n $ conditioned on eventual extinction.

For part two, I have seen that the probability of eventual extension, $\displaystyle \beta$, is given by

$\displaystyle \beta = I_{\{p \leq \frac{1}{2} \}} + \alpha I_{\{p > \frac{1}{2}\}} \ \mbox{where} \ \alpha = \frac{q}{p} $