# Beta prime distribution

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

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

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

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

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