Past exam of the options II for mathematics and philosophy course of the University of Oxford / 2013 / N2 / e / ii / Solution

For $|s|\le 1$, all the generating-function expectations are absolutely bounded. Conditional on $X=k$, independence givesAt $k=0$ the product is one, consistent with the empty-family case. The law of total expectation now yieldsThis is the total progeny generating function of a branching process. The given almost-sure finiteness of $T$ ensures $G_T(1)=1$. The conditional-expectation proof avoids assuming a finite expected total before it has been established.