# betainc

Incomplete beta function

## Syntax

I = betainc(X,Z,W)
I = betainc(X,Z,W,tail)

## Description

I = betainc(X,Z,W) computes the incomplete beta function for corresponding elements of the arrays X, Z, and W. The elements of X must be in the closed interval [0,1]. The arrays Z and W must be nonnegative and real. All arrays must be the same size, or any of them can be scalar.

I = betainc(X,Z,W,tail) specifies the tail of the incomplete beta function. Choices are:

 'lower' (the default) Computes the integral from 0 to x 'upper' Computes the integral from x to 1

These functions are related as follows:

1-betainc(X,Z,W) = betainc(X,Z,W,'upper')
Note that especially when the upper tail value is close to 0, it is more accurate to use the 'upper' option than to subtract the 'lower' value from 1.

## Examples

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Compute the incomplete beta function corresponding to the elements of Z according to the parameters X and W.

format longG
X = 0.5;
Z = (1:10)';
W = 3;
I = betainc(X,Z,W)
I = 10×1

0.8750
0.6875
0.5000
0.3438
0.2266
0.1445
0.0898
0.0547
0.0327
0.0193

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### Incomplete Beta Function

The incomplete beta function is

${I}_{x}\left(z,w\right)=\frac{1}{B\left(z,w\right)}{\int }_{0}^{x}{t}^{z-1}{\left(1-t\right)}^{w-1}dt$

where $B\left(z,w\right)$, the beta function, is defined as

$B\left(z,w\right)={\int }_{0}^{1}{t}^{z-1}{\left(1-t\right)}^{w-1}dt=\frac{\Gamma \left(z\right)\Gamma \left(w\right)}{\Gamma \left(z+w\right)}$

and $\Gamma \left(z\right)$ is the gamma function.