Past exam of the year 1 of the computer science course of the University of Oxford / 2016 / Discrete mathematics, continuous mathematics and probability / 8 / b / i / Solution

For a nonnegative integer-valued random variable $X$, its probability generating function isAt $s=0$, use $0^0=1$ so $G_X(0)=\mathbb{P}(X=0)$. The series also converges absolutely for complex $|s|\le 1$; for $s$ near zero, its coefficients recover the probability mass function.