Beta prime distribution

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A Beta Prime Distribution is a probability distribution defined for x>0 with two parameters (of positive real part), α and β, having the probability density function:

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

where $B$ is a Beta function. It is basically the same as the F distribution--if b is distributed as the beta prime distribution Beta'(α,β), then bβ/α obeys the F distribution with 2α and 2β degrees of freedom.

The mode of a variate $X$ distributed as $β' (α,β)$ is $\hat{X} = \frac{\alpha-1}{\beta+1}$ . Its mean is $\frac{\alpha}{\beta-1}$ and its variance is $\frac{\alpha(\alpha+\beta-1)}{(\beta-2)(\beta-1)^2}$ .

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.

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MathWorld article

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Beta prime distribution

From Wikipedia, the free encyclopedia

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