Noncentral beta distribution

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

Noncentral Beta

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

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

${\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2}}},}$

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

A Type II noncentral beta distribution is the distribution of the ratio

${\displaystyle Y={\frac {\chi _{n}^{2}}{\chi _{n}^{2}+\chi _{m}^{2}(\lambda )}},}$

where the noncentral chi-squared variable is in the denominator only. If ${\displaystyle Y}$ follows the type II distribution, then ${\displaystyle X=1-Y}$ follows a type I distribution.