Noncentral beta distribution information

Noncentral Beta
Notation	Beta(α, β, λ)
Parameters	α > 0 shape (real); β > 0 shape (real); λ ≥ 0 noncentrality (real)
Support
PDF	(type I)
CDF	(type I)
Mean	(type I) (see Confluent hypergeometric function)
Variance	(type I) where is the mean. (see Confluent hypergeometric function)

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

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 $\chi _{m}^{2}(\lambda )$ .^[1] 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.^[1] If $Y$ follows the type II distribution, then $X=1-Y$ follows a type I distribution.

^ ^a ^b Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151.

[Chat-1] Chattamvelli, R. (1995). "A Note on the Noncentral Beta Distribution Function". The American Statistician. 49 (2): 231–234. doi:10.1080/00031305.1995.10476151.

Noncentral beta distribution information

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Urn problem

Kelly criterion

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