= Inverse-gamma distribution
{wiki}
The <inverse-gamma distribution> with shape $a>0$ and scale $b>0$ has <probability density function>
$$
f_X(x)=\frac{b^a}{\Gamma(a)}x^{-a-1}e^{-b/x},\qquad x>0.
$$
Equivalently, $Y=1/X$ has a <gamma distribution> with shape $a$ and \i[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 $\mathbb EX^r=b^r\Gamma(a-r)/\Gamma(a)$ when $r<a$. For positive $r\geq a$ the defining <integral> diverges at infinity. In particular,
$$
\mathbb EX=\frac{b}{a-1}\quad(a>1),\qquad
\operatorname{Var}(X)=\frac{b^2}{(a-1)^2(a-2)}\quad(a>2).
$$
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>.
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