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Encyclopedia > Beta prime distribution

A Beta Prime Distribution is a distribution with probability function:

$P(x) = frac{x^(alpha-1) (1+x)^{-alpha -beta}}{B(alpha,beta)}$

where $B$ is a Beta function.

The mode of a variate $X$ distributed as $β' (α,β)$ is $hat{X} = frac{alpha-1}{beta+1}$ .

If X is a $β' (α,β)$ variate then $frac{1}{X}$ is a $β' (β,α)$ variate.

If X is a $β' (α,β)$ then $frac{1-X}{X}$ and $frac{X}{1-X}$ are $β' (β,α)$ and $β' (α,β)$ variates.

If X and Y are $γ(α 1)$ and $γ(α 2)$ variates, then $frac{X}{Y}$ is a $β' (α 1,α 2)$ variate.

If $frac{X^2}{2}$ and $frac{Y^2}{2}$ are $gamma(frac{1}{2})$ variates, then $Z^2 = left(frac{X}{Y}right)^2$ is a $beta^{'}(frac{1}{2},frac{1}{2})$ variate.

M.E. - November 2005

Results from FactBites:

probability distribution: Definition and Much More from Answers.com (1593 words)
	The Beta distribution on [0,1], of which the uniform distribution is a special case, and which is useful in estimating success probabilities.
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Beta distribution - Wikipedia, the free encyclopedia (597 words)
	The Dirichlet distribution is the multivariate generalization of the beta distribution.
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