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  "body": "\n## Summary\n\nThe beta function $B(x, y)$ is the integral $\\int_0^1 t^{x-1}(1 - t)^{y-1}\\, dt$ for positive $x$ and $y$.\nIt is expressed through the [[gamma function|0V3FZMVB4K]] by $B(x, y) = \\Gamma(x)\\Gamma(y)/\\Gamma(x + y)$.\nIt turns many integrals over intervals and over angles into products of gamma values.\nIt also gives the normalising constants of the beta distributions in probability.\nAt positive integers it is the reciprocal of a multiple of a binomial coefficient.\n\n## Explanations\n\n### L1 human\n\nThe beta function measures the area under the curve $t^{x-1}(1-t)^{y-1}$ between $0$ and $1$.\nLarge $x$ pushes the mass of this curve toward $t = 1$, and large $y$ pushes it toward $t = 0$.\nFor $x = y = 1$ the curve is the constant $1$ and the area is $1$.\nFor $x = 2$, $y = 1$ the curve is $t$ and the area is $\\frac12$.\n",
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