Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Discrete mathematics, probability and continuous mathematics / 8 / b / ii

The probability generating function of an independent random sum states that, for a nonnegative integer-valued $N$ independent of independent and identically distributed nonnegative integer-valued summands, $G_{{\sum }_{i=1}^N{X}_i}(s)=G_N(G_{X_1}(s))$ wherever the expectations converge. Here $G_{X_1}(s)=1-r+rs$, so, writing $q=1-p$,It is valid at least for $|s|\le 1$, and its power series in fact converges for $|s|<(p+qr)/(qr)$.