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  "body": "\n## Statement\n\nLet $X_1, \\dots, X_n$ be [[independent|0XCE070V8B]] real random variables, and let $S = X_1 + \\dots + X_n$.\nThen for each $t \\in \\mathbb{R}$, the [[moment-generating functions|0NNRNGKMFM]] satisfy\n$$M_S(t) = M_{X_1}(t) \\, M_{X_2}(t) \\cdots M_{X_n}(t),$$\nin $(0, \\infty]$.\n\n## Remarks\n\nProof: $e^{tS} = e^{tX_1} \\cdots e^{tX_n}$. The factors are nonnegative and independent, so the [[product rule for expectations|05AW1G0WE1]] applies.\nFor i.i.d. $X_i$, $M_S(t) = M_{X_1}(t)^n$.\nExample: if $X_i \\sim \\operatorname{Exp}(\\lambda)$ are independent, then $M_S(t) = (\\lambda/(\\lambda - t))^n$ for $t \u003c \\lambda$. This is the moment-generating function of the gamma distribution $\\operatorname{Gamma}(n, \\lambda)$.\n\n## Explanations\n\n### L1 human\n\nWhen you add independent random variables, their moment-generating functions multiply.\nThis makes it easy to identify the law of a sum, if you know which law has the product as its moment-generating function.\n",
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