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  "body": "\n## Definition\n\nLet $X$ be a real random variable with [[moment-generating function|0NNRNGKMFM]] $M_X$.\nThe cumulant generating function of $X$ is\n$$K_X(t) = \\ln M_X(t) = \\ln E\\left[e^{tX}\\right],$$\nfor each $t \\in \\mathbb{R}$ with $M_X(t) \u003c \\infty$.\nIf $M_X$ is finite on an open interval around $0$, the cumulants $\\kappa_n$ of $X$ are the coefficients in $K_X(t) = \\sum_{n=1}^{\\infty} \\kappa_n t^n / n!$, so $\\kappa_n = K_X^{(n)}(0)$.\n\n## Remarks\n\nThe first cumulants are $\\kappa_1 = E[X]$, $\\kappa_2 = \\operatorname{Var}(X)$ and $\\kappa_3 = E[(X - E[X])^3]$.\nFor independent $X$ and $Y$, $K_{X+Y} = K_X + K_Y$ by the [[product rule for moment-generating functions|0WRZ170YEH]]. So cumulants of independent sums add.\nFor $X \\sim \\mathcal{N}(\\mu, \\sigma^2)$, $K_X(t) = \\mu t + \\sigma^2 t^2 / 2$. For $X \\sim \\operatorname{Poisson}(\\lambda)$, $K_X(t) = \\lambda(e^t - 1)$, so all cumulants equal $\\lambda$.\n$K_X$ is convex on the interval where it is finite.\n\n## Explanations\n\n### L1 human\n\nTaking the logarithm turns products into sums. So the cumulant generating function of a sum of independent variables is the sum of their cumulant generating functions.\nThis makes cumulants convenient for sums, for example in the central limit theorem.\n",
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