# Inverse-gamma distribution

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Parameters Probability density function325px Cumulative distribution function325px ${\displaystyle \alpha >0}$ shape (real)${\displaystyle \beta >0}$ scale (real) ${\displaystyle x\in (0;\infty )\!}$ ${\displaystyle {\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{-\alpha -1}\exp \left({\frac {-\beta }{x}}\right)}$ ${\displaystyle {\frac {\Gamma (\alpha ,\beta /x)}{\Gamma (\alpha )}}\!}$ ${\displaystyle {\frac {\beta }{\alpha -1}}\!}$ for ${\displaystyle \alpha >1}$ ${\displaystyle {\frac {\beta }{\alpha +1}}\!}$ ${\displaystyle {\frac {\beta ^{2}}{(\alpha -1)^{2}(\alpha -2)}}\!}$ for ${\displaystyle \alpha >2}$ ${\displaystyle {\frac {4{\sqrt {\alpha -2}}}{\alpha -3}}\!}$ for ${\displaystyle \alpha >3}$ ${\displaystyle {\frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)}}\!}$ for ${\displaystyle \alpha >4}$ ${\displaystyle \alpha \!+\!\ln(\beta \Gamma (\alpha ))\!-\!(1\!+\!\alpha )\psi (\alpha )}$ ${\displaystyle {\frac {2\left(-\beta t\right)^{\!\!{\frac {\alpha }{2}}}}{\Gamma (\alpha )}}K_{\alpha }\left({\sqrt {-4\beta t}}\right)}$ ${\displaystyle {\frac {2\left(-i\beta t\right)^{\!\!{\frac {\alpha }{2}}}}{\Gamma (\alpha )}}K_{\alpha }\left({\sqrt {-4i\beta t}}\right)}$

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.

## Characterization

### Probability density function

The inverse gamma distribution's probability density function is defined over the support ${\displaystyle x>0}$

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

with shape parameter ${\displaystyle \alpha }$ and scale parameter ${\displaystyle \beta }$.

### Cumulative distribution function

The cumulative distribution function is the regularized gamma function

${\displaystyle F(x;\alpha ,\beta )={\frac {\Gamma (\alpha ,\beta /x)}{\Gamma (\alpha )}}\!}$

where the numerator is the upper incomplete gamma function and the denominator is the gamma function.

## Related distributions

• If ${\displaystyle X\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )}$ and ${\displaystyle \alpha ={\frac {\nu }{2}}}$ and ${\displaystyle \beta ={\frac {1}{2}}}$ then ${\displaystyle X\sim {\mbox{Inv-chi-square}}(\nu )\,}$ is an inverse-chi-square distribution
• If ${\displaystyle X\sim {\mbox{Inv-Gamma}}(k,\theta )\,}$ , then ${\displaystyle 1/X\sim {\mbox{Gamma}}(k,\theta ^{-1})\,}$ is a Gamma distribution
• A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.

## Derivation from Gamma distribution

The pdf of the gamma distribution is

${\displaystyle f(x)=x^{k-1}{\frac {e^{-x/\theta }}{\theta ^{k}\,\Gamma (k)}}}$

and define the transformation ${\displaystyle Y=g(X)={\frac {1}{X}}}$ then the resulting transformation is

${\displaystyle f_{Y}(y)=f_{X}\left(g^{-1}(y)\right)\left|{\frac {d}{dy}}g^{-1}(y)\right|}$
${\displaystyle ={\frac {1}{\theta ^{k}\Gamma (k)}}\left({\frac {1}{y}}\right)^{k-1}\exp \left({\frac {-1}{\theta y}}\right){\frac {1}{y^{2}}}}$
${\displaystyle ={\frac {1}{\theta ^{k}\Gamma (k)}}\left({\frac {1}{y}}\right)^{k+1}\exp \left({\frac {-1}{\theta y}}\right)}$
${\displaystyle ={\frac {1}{\theta ^{k}\Gamma (k)}}y^{-k-1}\exp \left({\frac {-1}{\theta y}}\right)}$

Replacing ${\displaystyle k}$ with ${\displaystyle \alpha }$; ${\displaystyle \theta ^{-1}}$ with ${\displaystyle \beta }$; and ${\displaystyle y}$ with ${\displaystyle x}$ results in the inverse-gamma pdf shown above

${\displaystyle f(x)={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}x^{-\alpha -1}\exp \left({\frac {-\beta }{x}}\right)}$