Noncentral BetaNotation |
Beta(α, β, λ) |
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Parameters |
α > 0 shape (real) β > 0 shape (real) λ ≥ 0 noncentrality (real) |
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Support |
![{\displaystyle x\in [0;1]\!}](https://wikimedia.org/api/rest_v1/media/math/render/svg/394f69db847ba283727b0bc73bccc019572a72ae) |
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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)))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e2f416e3014c4f6d7feb383e1a675cf5a350a8f5) |
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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)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c5f69f89b3478df9b2976844884e77793af99eb) |
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Mean |
(type I) (see Confluent hypergeometric function) |
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Variance |
(type I) where is the mean. (see Confluent hypergeometric function) |
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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
![{\displaystyle X={\frac {\chi _{m}^{2}(\lambda )}{\chi _{m}^{2}(\lambda )+\chi _{n}^{2))},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e92c23ac937619950544b8dc06a574a4b1e495a)
where
is a
noncentral chi-squared random variable with degrees of freedom m and noncentrality parameter
, and
is a central chi-squared random variable with degrees of freedom n, independent of
.[1]
In this case,
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 ))),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/981969faaa19262fa1725a4d7ff65893b9cacb5c)
where the noncentral chi-squared variable is in the denominator only.[1] If
follows
the type II distribution, then
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:[1]
![{\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha +j,\beta ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70592a41464846759f7fb20c4404b586e61594f8)
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
is the incomplete beta function. That is,
![{\displaystyle F(x)=\sum _{j=0}^{\infty }{\frac {1}{j!))\left({\frac {\lambda }{2))\right)^{j}e^{-\lambda /2}I_{x}(\alpha +j,\beta ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29156d2a96d601e354fa39a8f07b0bf3e67aabd3)
The Type II cumulative distribution function in mixture form is
![{\displaystyle F(x)=\sum _{j=0}^{\infty }P(j)I_{x}(\alpha ,\beta +j).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6f45511e4124ccdf110d8df2d7d1f45263f8eb9a)
Algorithms for evaluating the noncentral beta distribution functions are given by Posten[2] and Chattamvelli.[1]
Related distributions
Transformations
If
, then
follows a noncentral F-distribution with
degrees of freedom, and non-centrality parameter
.
If
follows a noncentral F-distribution
with
numerator degrees of freedom and
denominator degrees of freedom, then
![{\displaystyle Z={\cfrac {\cfrac {\mu _{2)){\mu _{1))}((\cfrac {\mu _{2)){\mu _{1))}+X^{-1))))](https://wikimedia.org/api/rest_v1/media/math/render/svg/e56bea7b737b20ed5ca164e6aa59931bbaf144d1)
follows a noncentral Beta distribution:
.
This is derived from making a straightforward transformation.
Special cases
When
, the noncentral beta distribution is equivalent to the (central) beta distribution.