Inverse-chi-square distribution

The inverse-chi-square distribution arises in Bayesian data analysis (spam filtering in particular). Its probability density function over the domain $x > 0$ is

$p(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)} \cdot x^{-\nu/2+1} \cdot e^{-1/(2 x)}$

Related distributions

chi-square relation with the inverse-chi-square is $Y = \frac{\nu}{X}$ where $X \sim \chi^2(x; \nu)$ and $Y ~ \sim \mbox{Inv-}\chi^2(\nu)$
scale-inverse-chi-square with $σ 2 = 1$
Inverse gamma with $\alpha = \frac{\nu}{2}$ and $\beta = \frac{1}{2}$