# Inverse-gamma distribution

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

### Cumulative distribution function

The cumulative distribution function is the regularized gamma function

$F(x;\alpha ,\beta )={\frac {\Gamma (\alpha ,\beta /x)}{\Gamma (\alpha )}}\!$ where the numerator is the upper incomplete gamma function and the denominator is the gamma function.

## Related distributions

• If $X\sim {\mbox{Inv-Gamma}}(\alpha ,\beta )$ and $\alpha ={\frac {\nu }{2}}$ and $\beta ={\frac {1}{2}}$ then $X\sim {\mbox{Inv-chi-square}}(\nu )\,$ is an inverse-chi-square distribution
• If $X\sim {\mbox{Inv-Gamma}}(k,\theta )\,$ , then $1/X\sim {\mbox{Gamma}}(k,\theta ^{-1})\,$ 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

$f(x)=x^{k-1}{\frac {e^{-x/\theta }}{\theta ^{k}\,\Gamma (k)}}$ and define the transformation $Y=g(X)={\frac {1}{X}}$ then the resulting transformation is

$f_{Y}(y)=f_{X}\left(g^{-1}(y)\right)\left|{\frac {d}{dy}}g^{-1}(y)\right|$ $={\frac {1}{\theta ^{k}\Gamma (k)}}\left({\frac {1}{y}}\right)^{k-1}\exp \left({\frac {-1}{\theta y}}\right){\frac {1}{y^{2}}}$ $={\frac {1}{\theta ^{k}\Gamma (k)}}\left({\frac {1}{y}}\right)^{k+1}\exp \left({\frac {-1}{\theta y}}\right)$ $={\frac {1}{\theta ^{k}\Gamma (k)}}y^{-k-1}\exp \left({\frac {-1}{\theta y}}\right)$ Replacing $k$ with $\alpha$ ; $\theta ^{-1}$ with $\beta$ ; and $y$ with $x$ results in the inverse-gamma pdf shown above

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