{
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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] random variables with values in $\\{0, 1, 2, \\dots\\}$, and let $S = X_1 + \\dots + X_n$.\nThen for each complex $z$ with $|z| \\le 1$, the [[probability-generating functions|0Z8PSSAVBA]] satisfy\n$$G_S(z) = G_{X_1}(z) \\, G_{X_2}(z) \\cdots G_{X_n}(z).$$\n\n## Remarks\n\nProof: $z^S = z^{X_1} \\cdots z^{X_n}$. The factors are independent, so the [[product rule for expectations|05AW1G0WE1]] applies.\nExample: a sum of $n$ independent Bernoulli($p$) variables has the generating function $(1 - p + pz)^n$. This is the generating function of $\\operatorname{Bin}(n, p)$.\nFor a random number $N$ of i.i.d. terms $X_i$ with $N$ independent of the $X_i$, the sum $S_N = X_1 + \\dots + X_N$ has $G_{S_N}(z) = G_N(G_{X_1}(z))$.\n\n## Explanations\n\n### L1 human\n\nAdding independent counts corresponds to multiplying their generating functions.\nThis turns the hard computation of the law of a sum into a product of power series.\n",
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  "msc": [
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  "name": "Generating function of a sum of independent random variables",
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