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  "body": "\n## Definition\n\nLet $X$ be a real random variable.\nThe moment-generating function of $X$ is the function\n$$M_X(t) = E\\left[e^{tX}\\right], \\quad t \\in \\mathbb{R},$$\nwith values in $(0, \\infty]$.\n\n## Remarks\n\nAlways $M_X(0) = 1$. The set of $t$ with $M_X(t) \u003c \\infty$ is an interval that contains $0$.\nIf $M_X$ is finite on an open interval around $0$, then all [[moments|0PH0VHTQS0]] of $X$ are finite and $M_X$ determines the law of $X$.\nSome random variables have $M_X(t) = \\infty$ for all $t \\ne 0$. An example is a Cauchy random variable.\nFor $X \\sim \\operatorname{Exp}(\\lambda)$, $M_X(t) = \\lambda / (\\lambda - t)$ for $t \u003c \\lambda$, and $M_X(t) = \\infty$ for $t \\ge \\lambda$.\n\n## Explanations\n\n### L1 human\n\nThe moment-generating function is a tool that turns a distribution into a function of one real variable.\nIt is useful because sums of independent variables become products, and moments become derivatives.\n",
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