Past exam of the year 1 of the computer science course of the University of Oxford / 2022 / Discrete mathematics, continuous mathematics, and probability / Probability / 8 / a / i / Solution

For a nonnegative integer-valued random variable, its probability generating function isThe series converges absolutely at least for $|s|\le 1$, with $0^0=1$ in the $k=0$ term; it may converge on a larger domain.