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  "body": "\n## Summary\n\nThe incomplete beta function $B(x; a, b)$ is the integral of $t^{a-1}(1-t)^{b-1}$ from $0$ to $x$, for $0 \\le x \\le 1$.\nAt $x = 1$ it is the [[beta function|QHNWVSC6AQ]] $B(a, b)$.\nDivided by $B(a, b)$ it becomes the regularised incomplete beta function $I_x(a, b)$, the distribution function of the [[beta distribution|0EW824Z7G2]].\nIt also gives the distribution functions of the binomial distribution and of Student's $t$-distribution.\n\n## Explanations\n\n### L1 human\n\nThe beta function is the area under the curve $t^{a-1}(1-t)^{b-1}$ between $0$ and $1$.\nThe incomplete beta function keeps only the part of this area to the left of a point $x$.\nAs $x$ moves from $0$ to $1$, it grows from $0$ to the full beta function.\nFor $a = b = 1$ the curve is the constant $1$, and the area up to $x$ is just $x$.\n",
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