Inverse-gamma distribution

The inverse gamma distribution has the probability density function over the support $x > 0$

$p(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left( \frac{-\beta}{x} \right)$

with shape parameter $α$ and scale parameter $β$ .

Related distributions

$X \sim \mbox{Inv-Gamma}(\alpha, \beta)$ then $Y \sim \mbox{Inv-chi-square}(\nu)$ if $\alpha = \frac{\nu}{2}$ and $\beta = \frac{1}{2}$ ; $Y$ is the Inverse-chi-square distribution
$X \sim \mbox{Gamma}(k, \theta)$ then $Y \sim \mbox{Inv-Gamma}(k, \theta)$ if $Y = \frac{1}{X}$ ; $X$ is the Gamma distribution