 Solution

The probability generating function of a nonnegative integer-valued random variable isUse $s^0=1$, including at $s=0$. The power series converges absolutely for every real $s\in [-1,1]$, since ${\sum }_{j}p(j)|s{|}^j\le 1$. In particular it is always defined on the usual probability-generating interval $[0,1]$. It may also be defined for $|s|>1$, depending on the radius of convergence and convergence at its endpoints; the precise real domain of finite expectation is $\{s\in \mathbb{R}:{\sum }_{j}p(j)|s{|}^j<\infty \}$.