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  "body": "\n## Definition\n\nLet $a \u003e 0$ and $b \u003e 0$ be real numbers and let $0 \\le x \\le 1$.\nThe *incomplete beta function* is\n$$B(x; a, b) = \\int_0^x t^{a-1} (1-t)^{b-1}\\, dt.$$\nThe *regularised incomplete beta function* is $I_x(a, b) = B(x; a, b)/B(a, b)$, where $B(a, b) = B(1; a, b)$ is the [[beta function|QHNWVSC6AQ]].\n\n## Remarks\n\nExample: for $a = 2$ and $b = 1$, $B(x; 2, 1) = \\int_0^x t\\, dt = \\frac{x^2}{2}$ and $B(2, 1) = \\frac12$, so $I_x(2, 1) = x^2$.\n\nNon-example of a valid argument: for $a = 0$ the integrand is $t^{-1}(1-t)^{b-1}$, whose integral diverges at $0$, so $B(x; 0, b)$ is not defined for $x \u003e 0$.\n\nThe function $x \\mapsto I_x(a, b)$ increases from $0$ to $1$ and is the distribution function of the [[beta distribution|0EW824Z7G2]] with parameters $a$ and $b$.\nThe substitution $t \\mapsto 1 - t$ gives the symmetry $I_x(a, b) = 1 - I_{1-x}(b, a)$.\nFor positive integers $a$ and $n$ with $a \\le n$, $I_p(a, n - a + 1)$ is the probability of at least $a$ successes in $n$ independent trials with success probability $p$.\n",
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