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  "body": "\n## Definition\n\nLet $\\alpha \u003e 0$ and $\\beta \u003e 0$.\nThe beta distribution with parameters $\\alpha$ and $\\beta$ is the distribution on $\\mathbb{R}$ with the [[probability density function|04YG690N5V]]\n$$f(x) = \\begin{cases} \\dfrac{x^{\\alpha - 1} (1 - x)^{\\beta - 1}}{B(\\alpha, \\beta)} \u0026 \\text{if } 0 \u003c x \u003c 1, \\\\ 0 \u0026 \\text{otherwise,} \\end{cases}$$\nwhere $B(\\alpha, \\beta) = \\int_0^1 t^{\\alpha - 1} (1 - t)^{\\beta - 1} \\, dt$ is the beta function.\n\n## Remarks\n\nThe mean is $\\alpha / (\\alpha + \\beta)$ and the variance is $\\dfrac{\\alpha\\beta}{(\\alpha + \\beta)^2 (\\alpha + \\beta + 1)}$.\n$\\operatorname{Beta}(1, 1)$ is the [[uniform distribution|0WGDPYR30Q]] on $[0, 1]$.\nThe beta function satisfies $B(\\alpha, \\beta) = \\Gamma(\\alpha)\\Gamma(\\beta)/\\Gamma(\\alpha + \\beta)$.\nIn Bayesian statistics, the beta distribution is the conjugate prior of the success probability $p$ of binomial data. A $\\operatorname{Beta}(\\alpha, \\beta)$ prior and $k$ successes in $n$ trials give a $\\operatorname{Beta}(\\alpha + k, \\beta + n - k)$ posterior.\n\n## Explanations\n\n### L1 human\n\nThe beta distribution describes a random proportion, a number between $0$ and $1$.\nLarge $\\alpha$ pushes the mass toward $1$, large $\\beta$ pushes it toward $0$, and $\\alpha = \\beta$ gives a shape that is symmetric about $1/2$.\n",
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