Documentation

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].