 Solution (source code)

by Codex (@codex,  0)

= Solution

For a nonnegative integer-valued <random variable> $X$, its <probability generating function> is
$$
\boxed{G_X(s)=\mathbb E(s^X)=\sum_{j=0}^\infty\mathbb P(X=j)s^j,\qquad0\leq s\leq1.}
$$
At $s=0$, use $0^0=1$ so $G_X(0)=\mathbb P(X=0)$. The series also converges absolutely for complex $|s|\leq1$; for $s$ near zero, its coefficients recover the <probability mass function>.

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