Inverse gamma distribution

Inverse-gamma

Probability density function
Cumulative distribution function
Parameters	${\displaystyle \alpha >0}$ shape (real) ${\displaystyle \beta >0}$ rate (real)
Support	${\displaystyle x\in (0,\infty )\!}$
PDF	${\displaystyle {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{-\alpha -1}\exp \left(-{\frac {\beta }{x}}\right)}$
CDF	${\displaystyle {\frac {\Gamma (\alpha ,\beta /x)}{\Gamma (\alpha )}}\!}$
Mean	${\displaystyle {\frac {\beta }{\alpha -1}}\!}$ for ${\displaystyle \alpha >1}$
Mode	${\displaystyle {\frac {\beta }{\alpha +1}}\!}$
Variance	${\displaystyle {\frac {\beta ^{2}}{(\alpha -1)^{2}(\alpha -2)}}\!}$ for ${\displaystyle \alpha >2}$
Skewness	${\displaystyle {\frac {4{\sqrt {\alpha -2}}}{\alpha -3}}\!}$ for ${\displaystyle \alpha >3}$
Ex. kurtosis	${\displaystyle {\frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)}}\!}$ for ${\displaystyle \alpha >4}$
Entropy	${\displaystyle \alpha \!+\!\ln(\beta \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\Psi (\alpha )}$ (see digamma function)
MGF	Does not exist.
CF	${\displaystyle {\frac {2\left(-i\beta t\right)^{\!\!{\frac {\alpha }{2}}}}{\Gamma (\alpha )}}K_{\alpha }\left({\sqrt {-4i\beta t}}\right)}$

${\displaystyle \alpha \!+\!\ln(\beta \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\Psi (\alpha )}$

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required.