# Inverse-chi-square distribution

Parameters Probability density functionFile:Inverse chi squared density.png Cumulative distribution functionFile:Inverse chi squared distribution.png ${\displaystyle \nu >0\!}$ ${\displaystyle x\in (0,\infty )\!}$ ${\displaystyle {\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}\!}$ ${\displaystyle \Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /}\,\Gamma \!\left({\frac {\nu }{2}}\right)\!}$ ${\displaystyle {\frac {1}{\nu -2}}\!}$ for ${\displaystyle \nu >2\!}$ ${\displaystyle {\frac {1}{\nu +2}}\!}$ ${\displaystyle {\frac {2}{(\nu -2)^{2}(\nu -4)}}\!}$ for ${\displaystyle \nu >4\!}$ ${\displaystyle {\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}\!}$ for ${\displaystyle \nu >6\!}$ ${\displaystyle {\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}\!}$ for ${\displaystyle \nu >8\!}$ ${\displaystyle {\frac {\nu }{2}}\!+\!\ln \!\left({\frac {1}{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)}$ ${\displaystyle \!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)}$ ${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)}$ ${\displaystyle {\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)}$

In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-square distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-square distribution. That is, if ${\displaystyle X}$ has the chi-square distribution with ${\displaystyle \nu }$ degrees of freedom, then according to the first definition, ${\displaystyle 1/X}$ has the inverse-chi-square distribution with ${\displaystyle \nu }$ degrees of freedom; while according to the second definition, ${\displaystyle \nu /X}$ has the inverse-chi-square distribution with ${\displaystyle \nu }$ degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function

${\displaystyle f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}}$

The second definition yields a density function

${\displaystyle f(x;\nu )={\frac {(\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}x^{-\nu /2-1}e^{-\nu /(2x)}}$

In both cases, ${\displaystyle x>0}$ and ${\displaystyle \nu }$ is the degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scale-inverse-chi-square distribution. For the first definition ${\displaystyle \sigma ^{2}=1/\nu }$ and for the second definition ${\displaystyle \sigma ^{2}=1}$.

## Related distributions

• chi-square: If ${\displaystyle X\sim \chi ^{2}(\nu )}$ and ${\displaystyle Y={\frac {1}{X}}}$ then ${\displaystyle Y~\sim {\mbox{Inv-}}\chi ^{2}(\nu )}$.
• Inverse gamma with ${\displaystyle \alpha ={\frac {\nu }{2}}}$ and ${\displaystyle \beta ={\frac {1}{2}}}$