# Noncentral beta distribution

In probability theory and statistics, the noncentral beta distribution is a continuous probability distribution that is a generalization of the (central) beta distribution.

Notation Beta(α, β, λ) α > 0 shape (real)β > 0 shape (real)λ >= 0 noncentrality (real) $x\in [0;1]\!$ (type I) $\sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}{\frac {x^{\alpha +j-1}\left(1-x\right)^{\beta -1}}{\mathrm {B} \left(\alpha +j,\beta \right)}}$ (type I) $\sum _{j=0}^{\infty }e^{-\lambda /2}{\frac {\left({\frac {\lambda }{2}}\right)^{j}}{j!}}I_{x}\left(\alpha +j,\beta \right)$ (type I) $e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +1\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +1\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +1;\alpha ,\alpha +\beta +1;{\frac {\lambda }{2}}\right)$ (see Confluent hypergeometric function) (type I) $e^{-{\frac {\lambda }{2}}}{\frac {\Gamma \left(\alpha +2\right)}{\Gamma \left(\alpha \right)}}{\frac {\Gamma \left(\alpha +\beta \right)}{\Gamma \left(\alpha +\beta +2\right)}}{}_{2}F_{2}\left(\alpha +\beta ,\alpha +2;\alpha ,\alpha +\beta +2;{\frac {\lambda }{2}}\right)-\mu ^{2}$ where $\mu$ is the mean. (see Confluent hypergeometric function)

The noncentral beta distribution (Type I) is the distribution of the ratio

$X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},$ where $\chi _{m}^{2}(\lambda )$ is a noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter $\lambda$ , and $\chi _{n}^{2}$ is a central chi-squared random variable with degrees of freedom n, independent of $\chi _{m}^{2}(\lambda )$ . In this case, $X\sim {\mbox{Beta}}\left({\frac {m}{2}},{\frac {n}{2}},\lambda \right)$ A Type II noncentral beta distribution is the distribution of the ratio

$Y={\frac {\chi _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},$ where the noncentral chi-squared variable is in the denominator only. If $Y$ follows the type II distribution, then $X=1-Y$ follows a type I distribution.

## Cumulative distribution function

The Type I cumulative distribution function is usually represented as a Poisson mixture of central beta random variables:

$F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),$ where λ is the noncentrality parameter, P(.) is the Poisson(λ/2) probability mass function, \alpha=m/2 and \beta=n/2 are shape parameters, and $I_{x}(a,b)$ is the incomplete beta function. That is,

$F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}I_{x}(\alpha +j,\beta ).$ The Type II cumulative distribution function in mixture form is

$F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha ,\beta +j).$ Algorithms for evaluating the noncentral beta distribution functions are given by Posten and Chattamvelli.

## Probability density function

The (Type I) probability density function for the noncentral beta distribution is:

$f(x)=\sum _{j=0}^{\infty }{\frac {1}{j!}}\left({\frac {\lambda }{2}}\right)^{j}e^{-\lambda /2}{\frac {x^{\alpha +j-1}(1-x)^{\beta -1}}{B(\alpha +j,\beta )}}.$ where $B$ is the beta function, $\alpha$ and $\beta$ are the shape parameters, and $\lambda$ is the noncentrality parameter. The density of Y is the same as that of 1-X with the degrees of freedom reversed.

### Transformations

If $X\sim {\mbox{Beta}}\left(\alpha ,\beta ,\lambda \right)$ , then ${\frac {\beta X}{\alpha (1-X)}}$ follows a noncentral F-distribution with $2\alpha ,2\beta$ degrees of freedom, and non-centrality parameter $\lambda$ .

If $X$ follows a noncentral F-distribution $F_{\mu _{1},\mu _{2}}\left(\lambda \right)$ with $\mu _{1}$ numerator degrees of freedom and $\mu _{2}$ denominator degrees of freedom, then $Z={\cfrac {\cfrac {\mu _{2}}{\mu _{1}}}{{\cfrac {\mu _{2}}{\mu _{1}}}+X^{-1}}}$ follows a noncentral Beta distribution so $Z\sim {\mbox{Beta}}\left({\frac {1}{2}}\mu _{1},{\frac {1}{2}}\mu _{2},\lambda \right)$ . This is derived from making a straightforward transformation.

### Special cases

When $\lambda =0$ , the noncentral beta distribution is equivalent to the (central) beta distribution.