# Documentation

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# betaincinv

Beta inverse cumulative distribution function

## Syntax

```x = betaincinv(y,z,w) x = betaincinv(y,z,w,tail) ```

## Description

`x = betaincinv(y,z,w)` computes the inverse incomplete beta function for corresponding elements of `y`, `z`, and `w`, such that `y = betainc(x,z,w)`. The elements of `y` must be in the closed interval [0,1], and those of `z` and `w` must be nonnegative. `y`, `z`, and `w` must all be real and the same size (or any of them can be scalar).

`x = betaincinv(y,z,w,tail)` specifies the tail of the incomplete beta function. Choices are `'lower'` (the default) to use the integral from 0 to `x`, or `'upper'` to use the integral from `x` to 1. These two choices are related as follows: `betaincinv(y,z,w,'upper') = betaincinv(1-y,z,w,'lower')`. When `y` is close to 0, the `'upper'` option provides a way to compute `x` more accurately than by subtracting `y` from 1.

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### Inverse Incomplete Beta Function

The incomplete beta function is defined as

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

`betaincinv` computes the inverse of the incomplete beta function with respect to the integration limit `x` using Newton's method.