Accelerating the pace of engineering and science

# betainc

Incomplete beta function

## Syntax

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

## Definitions

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.

## 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

```format long
betainc(.5,(0:10)',3)

ans =
1.00000000000000
0.87500000000000
0.68750000000000
0.50000000000000
0.34375000000000
0.22656250000000
0.14453125000000
0.08984375000000
0.05468750000000
0.03271484375000
0.01928710937500```