Noncentral beta distribution

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

Contents

1 Probability density function
2 Cumulative distribution function
3 Special cases
4 References

Probability density function

The probability density function for the noncentral beta distribution is:

$f(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}\frac{x^{a+j-1}(1-x)^{b-1}}{B(a+j,b)}$

where $B$ is the beta function, $a$ and $b$ are the shape parameters, and $\lambda$ is the noncentrality parameter.

Cumulative distribution function

The cumulative distribution function for the noncentral beta distribution is:

$F(x) = \sum_{j=0}^\infin \frac{1}{j!}\left(\frac{\lambda}{2}\right)^je^{-\lambda/2}I_x(a+j,b)$

where $I_x$ is the regularized incomplete beta function, $a$ and $b$ are the shape parameters, and $\lambda$ is the noncentrality parameter.

Special cases

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

This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please improve this article by introducing more precise citations. (August 2011)

References

M. Abramowitz and I. Stegun, editors (1965) "Handbook of Mathematical Functions", Dover: New York, NY.
Hodges, J.L. Jr (1955). "On the noncentral beta-distribution". Annals of Mathematical Statistics 26: 648–653.
Seber, G.A.F. (1963). "The non-central chi-squared and beta distributions". Biometrika 50: 542–544.

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Noncentral beta distribution