# Beta prime distribution

Parameters Probability density function325px Cumulative distribution function325px ${\displaystyle \alpha >0}$ shape (real)${\displaystyle \beta >0}$ shape (real) ${\displaystyle x>0\!}$ ${\displaystyle f(x)={\frac {x^{\alpha -1}(1+x)^{-\alpha -\beta }}{B(\alpha ,\beta )}}\!}$ ${\displaystyle {\frac {x^{\alpha }\cdot _{2}F_{1}(\alpha ,\alpha +\beta ,\alpha +1,-x)}{\alpha \cdot B(\alpha ,\beta )}}\!}$ where ${\displaystyle _{2}F_{1}}$ is the Gauss's hypergeometric function 2F1 ${\displaystyle {\frac {\alpha }{\beta -1}}}$ ${\displaystyle {\frac {\alpha -1}{\beta +1}}\!}$ ${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}$

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:

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

where ${\displaystyle B}$ 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 ${\displaystyle X}$ distributed as ${\displaystyle \beta ^{'}(\alpha ,\beta )}$ is ${\displaystyle {\hat {X}}={\frac {\alpha -1}{\beta +1}}}$. Its mean is ${\displaystyle {\frac {\alpha }{\beta -1}}}$ if ${\displaystyle \beta >1}$ (if ${\displaystyle \beta <=1}$ the mean is infinite, in other words it has no well defined mean) and its variance is ${\displaystyle {\frac {\alpha (\alpha +\beta -1)}{(\beta -2)(\beta -1)^{2}}}}$ if ${\displaystyle \beta >2}$.

If X is a ${\displaystyle \beta ^{'}(\alpha ,\beta )}$ variate then ${\displaystyle {\frac {1}{X}}}$ is a ${\displaystyle \beta ^{'}(\beta ,\alpha )}$ variate.

If X is a ${\displaystyle \beta (\alpha ,\beta )}$ then ${\displaystyle {\frac {1-X}{X}}}$ and ${\displaystyle {\frac {X}{1-X}}}$ are ${\displaystyle \beta ^{'}(\beta ,\alpha )}$ and ${\displaystyle \beta ^{'}(\alpha ,\beta )}$ variates.

If X and Y are ${\displaystyle \gamma (\alpha _{1})}$ and ${\displaystyle \gamma (\alpha _{2})}$ variates, then ${\displaystyle {\frac {X}{Y}}}$ is a ${\displaystyle \beta ^{'}(\alpha _{1},\alpha _{2})}$ variate.