Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics III / 5 / b / ii / Solution

For an independent nonnegative integer-valued count $N$ and identically distributed independent summands, the random sum probability generating function is $G_Z(s)=G_N(G_{X_1}(s))$. Conditioning on $N=n$ proves this, since the conditional probability generating function is $G_{X_1}(s{)}^n$.

Here $G_{X_1}(s)=1-r+rs$, soThis identity holds throughout $|s|\le 1$, and its power-series extension has radius $[p+(1-p)r]/[(1-p)r]$.