Beta prime distribution

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.