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  "body": "\n## Definition\n\nLet $X$ be a random variable with values in $\\{0, 1, 2, \\dots\\}$ and with [[mass function|0WF4EJP91W]] $p(k) = P(X = k)$.\nThe probability-generating function of $X$ is\n$$G_X(z) = E\\left[z^X\\right] = \\sum_{k=0}^{\\infty} p(k) \\, z^k,$$\nfor each complex $z$ with $|z| \\le 1$.\n\n## Remarks\n\n$G_X$ is the [[generating function|0HH6JX5WZZ]] of the sequence $p(0), p(1), \\dots$. The series converges absolutely for $|z| \\le 1$, and $G_X(1) = 1$.\n$G_X$ determines the law of $X$: $P(X = k) = G_X^{(k)}(0)/k!$.\nIf $E[X] \u003c \\infty$, then $E[X] = G_X'(1)$, with the left derivative at $1$. Also $E[X(X - 1)] = G_X''(1)$ when it is finite.\nFor $X \\sim \\operatorname{Poisson}(\\lambda)$, $G_X(z) = e^{\\lambda(z - 1)}$. For $X \\sim \\operatorname{Bin}(n, p)$, $G_X(z) = (1 - p + pz)^n$.\n\n## Explanations\n\n### L1 human\n\nThe probability-generating function packs all the probabilities of a count into one function.\nDerivatives of this function at $0$ give back the probabilities, and derivatives at $1$ give the moments.\n",
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