= Probability generating function
{title2=$G_X(s)=\mathbb E[s^X]$}
{wiki}
= PGF
{c}
{synonym}
For a nonnegative integer-valued <random variable> $X$, write $p_k=\mathbb P(X=k)$. Its <probability generating function> packages the entire <probability mass function> into one <power series>:
$$
G_X(s)=\mathbb E[s^X]=\sum_{k=0}^\infty p_ks^k,\qquad |s|\leq1.
$$
This domain is guaranteed, even for complex $s$, because $\sum_kp_k|s|^k\leq1$. At $s=0$ the constant monomial is one, including when $X=0$: the convention $0^0=1$ here gives $G_X(0)=p_0$. Also $G_X(1)=1$. The <power series> may converge beyond this domain, but that needs an additional argument. Encoding a <probability distribution> this way turns <convolution of independent random variables> into multiplication.<Random: Probability, Mathematical Statistics, Stochastic Processes by Kyle Siegrist>[Expected Value, §6 Generating Functions, Basic Theory, The Probability Generating Function: definition, absolute convergence, inversion, and the independent-sum identity]{ref}
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