# betainv

Beta inverse cumulative distribution function

## Syntax

```X = betainv(P,A,B) ```

## Description

`X = betainv(P,A,B)` computes the inverse of the beta cdf with parameters specified by `A` and `B` for the corresponding probabilities in `P`. `P`, `A`, and `B` can be vectors, matrices, or multidimensional arrays that are all the same size. A scalar input is expanded to a constant array with the same dimensions as the other inputs. The parameters in `A` and `B` must all be positive, and the values in `P` must lie on the interval [0, 1].

The inverse beta cdf for a given probability p and a given pair of parameters a and b is

`$x={F}^{-1}\left(p|a,b\right)=\left\{x:F\left(x|a,b\right)=p\right\}$`

where

`$p=F\left(x|a,b\right)=\frac{1}{B\left(a,b\right)}\underset{0}{\overset{x}{\int }}{t}^{a-1}{\left(1-t\right)}^{b-1}dt$`

and B( · ) is the Beta function. Each element of output `X` is the value whose cumulative probability under the beta cdf defined by the corresponding parameters in `A` and `B` is specified by the corresponding value in `P`.

## Examples

```p = [0.01 0.5 0.99]; x = betainv(p,10,5) x = 0.3726 0.6742 0.8981```

According to this result, for a beta cdf with a = 10 and b = 5, a value less than or equal to 0.3726 occurs with probability 0.01. Similarly, values less than or equal to 0.6742 and 0.8981 occur with respective probabilities 0.5 and 0.99.

## Algorithms

The `betainv` function uses Newton's method with modifications to constrain steps to the allowable range for x, i.e., [0 1].