Past exam of the year 1 of the computer science course of the University of Oxford / 2015 / Discrete mathematics, continuous mathematics, and probability (resit) / 8 / a / i / Solution

The probability generating function of a nonnegative integer-valued random variable isusing $s^0=1$ also at $s=0$. It is always defined for $0\le s\le 1$, and indeed the power series converges absolutely for all real or complex $s$ with $|s|\le 1$, becauseIt may be defined beyond the unit disk where this power series converges absolutely; such a larger domain depends on the distribution. In particular $G_X(0)=\mathbb{P}(X=0)$ and $G_X(1)=1$.