 Inverse-gamma distribution

The inverse-gamma distribution with shape $a>0$ and scale $b>0$ has probability density functionEquivalently, $Y=1/X$ has a gamma distribution with shape $a$ and rate $b$. The inverse change of variables has absolute Jacobian determinant $x^{-2}$, which turns the gamma density into the displayed expression. Thus the inverse-gamma scale parameter is the reciprocal variable's gamma rate, not its gamma scale.

Its moments satisfy $\mathbb{E}{X}^r=b^r\Gamma (a-r)/\Gamma (a)$ when $r<a$. For positive $r\ge a$ the defining integral diverges at infinity. In particular,For $a=3,b=2$, the expected value and variance are both one. The distribution is useful for positive scale uncertainty, but its heavy tail means that a valid normalized probability distribution can still lack a finite expected value.

The Inverse-Gamma distribution is a continuous probability distribution that is often used in Bayesian statistics, particularly in the context of prior distributions for variances. It is a two-parameter distribution that is defined over positive real numbers.