Beta Prime Distribution

A distribution with probability function

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

where $B$ is a Beta Function. The Mode of a variate distributed as $\beta'(\alpha, \beta)$ is

$$\hat x={\alpha-1\over \beta +1}.$$

If $x$ is a $\beta'(\alpha, \beta)$ variate, then $1/x$ is a $\beta'(\beta, \alpha)$ variate. If $x$ is a $\beta(\alpha,\beta)$ variate, then $(1-x)/x$ and $x/(1-x)$ are $\beta'(\beta, \alpha)$ and $\beta'(\alpha, \beta)$ variates. If $x$ and $y$ are $\gamma(\alpha_1)$ and $\gamma(\alpha_2)$ variates, then $x/y$ is a $\beta'(\alpha_1, \alpha_2)$ variate. If $x^2/2$ and $y^2/2$ are $\gamma(1/2)$ variates, then $z^2\equiv (x/y)^2$ is a $\beta '(1/2,1/2)$ variate.