BetaRegularized

The regularized incomplete beta function $I_x(a, b)$, the CDF of the Beta(a, b) distribution and a bridge to the binomial CDF.

✓ checked against that system · ! checked, with disagreements (hover a chip)Wolfram LanguageBetaRegularized!↗

BetaRegularized(x, a, b)the regularized incomplete beta function $I_x(a, b)$.

SymPybetainc_regularized(a, b, 0, x)Juliaenumeratio_ball((z, a, b) -> _betainc(z, a, b) / _beta(a, b), x, a, b)Rustbeta_regularized(a, b, x)

Typed signatures

Type	Package	Applies
(complex | infinity, complex | infinity, complex | infinity) -> number	compute-engine	always
(number, number, number, number?) -> number	analyticreplaces compute-engine	always

Domain: Special functions

Details

• Normalized against Beta: $I_x(a, b)$ is the incomplete beta integral at x, divided by $B(a, b)$, so $I_0(a,b)=0$ and $I_1(a,b)=1$.
• Symmetry: $I_x(a, b) + I_{1-x}(b, a) = 1$.
• Ties to the binomial CDF: for $X \sim \text{Binomial}(n, p)$, $P(X \leq k) = I_{1-p}(n-k, k+1)$.
• By symmetry, $I_{1/2}(a, a) = 1/2$ for any a: a symmetric Beta(a, a) distribution has its median exactly at the midpoint.
• Outside $[0, 1]$ the function continues analytically. At integers $a, b \ge 1$ it is a polynomial in x, the binomial tail $\sum_{j=a}^{n}\binom{n}{j}x^j(1-x)^{n-j}$ with $n = a+b-1$, so a rational x has an exact value ($I_2(2, 3) = 8$, $I_{-1}(2, 3) = 17$) and a float x the value at that double. A real $a > 0$ with integer b past $x = 1$ is summed in doubles. Anything else outside $[0, 1]$ (a non-integer b, where the value is complex) stays unevaluated.

Plot

Examples

1 more kept as data, checked against the oracles

Basic #

live ↗

In$\operatorname{BetaRegularized}(0, 2, 3)$Out$0$

live ↗

In$\operatorname{BetaRegularized}(1, 2, 3)$Out$1$

A floating-point x evaluates directly to a decimal

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In$\operatorname{BetaRegularized}(0.5, 2, 3)$Out$0.687\,5$

$I_x(1, 1) = x$, the uniform distribution's own CDF

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In$\operatorname{BetaRegularized}(0.5, 1, 1)$Out$0.5$

Non-integer parameters

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In$\operatorname{BetaRegularized}(0.4, 2.5, 3.5)$Out$0.486\,904\,191\,526\,117\,355\,254$

$I_{1/2}(2, 3) = 11/16$ exactly

live ↗

In$\operatorname{BetaRegularized}(\frac{1}{2}, 2, 3)$Out$\frac{11}{16}$

Scope #

Threads element-wise over a list, as Wolfram's Listable heads do

live ↗

In$\operatorname{BetaRegularized}(0.5, \bigl\lbrack1, 2\bigr\rbrack, 1)$Out$\bigl\lbrack0.5, 0.25\bigr\rbrack$

$x$ outside $[0, 1]$ by continuation: real here, as $b = 5$ makes the integrand a polynomial times $t^{-1/2}$

live ↗

In$\operatorname{BetaRegularized}(2, 0.5, 5)$Out$1.182\,194\,149\,796\,259\,1$

Four arguments: the generalized $I_{z_1}(a, b) - I_{z_0}(a, b)$

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In$\operatorname{BetaRegularized}(0.2, 0.5, 2, 3)$Out$0.506\,7$

Interval arithmetic: $I_x(2, 1) = x^2$

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In$\operatorname{BetaRegularized}(\mathrm{Interval}(0.2, 0.3), 2, 1)$Out$\mathrm{Interval}(0.04, 0.09)$

Uncertainty propagation

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In$\operatorname{BetaRegularized}(\mathrm{Around}(0.3, 0.01), 1, 2)$Out$\mathrm{Around}(0.51, 0.014)$

To 30 significant digits, every one correctly rounded

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In$N(\operatorname{BetaRegularized}(\frac{1}{3}, 2, 3), 30)$Out$0.407\,407\,407\,407\,407\,407\,407\,407\,407\,407$

Applications #

Binomial CDF via the regularized beta: for $X \sim \text{Binomial}(4, 0.3)$, $P(X \leq 1) = I_{0.7}(3, 2)$

live ↗

In$\operatorname{BetaRegularized}(0.7, 3, 2)=0.7^4+4×0.3×0.7^3$Out$\mathrm{True}$

Properties #

Symmetry: $I_x(a, b) + I_{1-x}(b, a) = 1$

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In$\operatorname{BetaRegularized}(0.3, 2, 3)+\operatorname{BetaRegularized}(0.7, 3, 2)=1$Out$\mathrm{True}$

$I_{1/2}(a, a) = 1/2$ for any a, here at $a = 10$

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In$\operatorname{BetaRegularized}(0.5, 10, 10)$Out$0.5$

$I_x(a, 1) = x^a$, here $0.2111\ldots^5$

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In$\operatorname{BetaRegularized}(0.211\,111\,111\,111\,111\,1, 5, 1)$Out$0.000\,419\,329\,539\,873\,664\,134\,599$

The arcsine distribution's median: $I_{1/2}(1/2, 1/2) = 1/2$

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In$\operatorname{BetaRegularized}(0.5, \frac{1}{2}, \frac{1}{2})$Out$0.5$

$I_x(1, 1) = x$ symbolically

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In$\operatorname{BetaRegularized}(x, 1, 1)$Out$x$

$I_x(a, 1) = x^a$ symbolically

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In$\operatorname{BetaRegularized}(x, a, 1)$Out$x^{a}$

$I_x(1, b) = 1 - (1-x)^b$

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In$\operatorname{BetaRegularized}(x, 1, b)$Out$1-(1-x)^{b}$

Possible issues #

Past $x = 1$ the function continues analytically: at integer $a$ and $b$ it is the polynomial $\sum_{j=a}^{a+b-1}\binom{a+b-1}{j}x^j(1-x)^{a+b-1-j}$

live ↗

In$\operatorname{BetaRegularized}(2, 2, 3)$Out$8$

About

Examples	22
Checked against	Wolfram Language 20 agree2 differSymPy 19 agree1 differJulia 15 agree1 errorRust 13 agree3 error
Crosswalk	Wolfram LanguageSymPyJuliaRust

Implementation

mappedsympypackages/oracle/src/mappings.ts

SymPy's betainc_regularized(a, b, x1, x2) is the integral between two limits, so the lower limit is 0 and the arguments reorder.

mappedjuliapackages/oracle/src/mappings.ts

mappedrustpackages/oracle/src/mappings.ts

The crate takes a, b, then the point.

See also: Beta, GammaRegularized, Binomial