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  "body": "\n## Definition\n\nLet $\\alpha \u003e 0$ and $\\beta \u003e 0$.\nThe gamma distribution with shape $\\alpha$ and rate $\\beta$ is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$f(x) = \\begin{cases} \\dfrac{\\beta^\\alpha}{\\Gamma(\\alpha)} x^{\\alpha - 1} e^{-\\beta x} \u0026 \\text{if } x \u003e 0, \\\\ 0 \u0026 \\text{if } x \\le 0, \\end{cases}$$\nwith the gamma function $\\Gamma(\\alpha) = \\int_0^\\infty t^{\\alpha - 1} e^{-t} \\, dt$. One writes $X \\sim \\operatorname{Gamma}(\\alpha, \\beta)$.\n\n## Remarks\n\nThe mean is $\\alpha / \\beta$ and the variance is $\\alpha / \\beta^2$. Another common form uses the scale $\\theta = 1/\\beta$.\n$\\operatorname{Gamma}(1, \\beta)$ is the [[exponential distribution|028Z8WC5Y9]] with rate $\\beta$. For an integer $n \\ge 1$, $\\operatorname{Gamma}(n, \\beta)$ is the law of a sum of $n$ independent $\\operatorname{Exp}(\\beta)$ random variables. It is the Erlang distribution.\nIf $X \\sim \\operatorname{Gamma}(\\alpha_1, \\beta)$ and $Y \\sim \\operatorname{Gamma}(\\alpha_2, \\beta)$ are independent, then $X + Y \\sim \\operatorname{Gamma}(\\alpha_1 + \\alpha_2, \\beta)$.\nThe chi-squared distribution with $k$ degrees of freedom is $\\operatorname{Gamma}(k/2, 1/2)$.\n\n## Explanations\n\n### L1 human\n\nThe gamma distribution with an integer shape $n$ is the waiting time for the $n$-th event of a Poisson process.\nFor example, it can model the time until the third customer arrives.\n",
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