# Inverse-chi-square distribution

Parameters Probability density functionFile:Inverse chi squared density.png Cumulative distribution functionFile:Inverse chi squared distribution.png $\nu >0\!$ $x\in (0,\infty )\!$ ${\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}\!$ $\Gamma \!\left({\frac {\nu }{2}},{\frac {1}{2x}}\right){\bigg /}\,\Gamma \!\left({\frac {\nu }{2}}\right)\!$ ${\frac {1}{\nu -2}}\!$ for $\nu >2\!$ ${\frac {1}{\nu +2}}\!$ ${\frac {2}{(\nu -2)^{2}(\nu -4)}}\!$ for $\nu >4\!$ ${\frac {4}{\nu -6}}{\sqrt {2(\nu -4)}}\!$ for $\nu >6\!$ ${\frac {12(5\nu -22)}{(\nu -6)(\nu -8)}}\!$ for $\nu >8\!$ ${\frac {\nu }{2}}\!+\!\ln \!\left({\frac {1}{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)$ $\!-\!\left(1\!+\!{\frac {\nu }{2}}\right)\psi \!\left({\frac {\nu }{2}}\right)$ ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-t}{2i}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2t}}\right)$ ${\frac {2}{\Gamma ({\frac {\nu }{2}})}}\left({\frac {-it}{2}}\right)^{\!\!{\frac {\nu }{4}}}K_{\frac {\nu }{2}}\!\left({\sqrt {-2it}}\right)$ 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 $X$ has the chi-square distribution with $\nu$ degrees of freedom, then according to the first definition, $1/X$ has the inverse-chi-square distribution with $\nu$ degrees of freedom; while according to the second definition, $\nu /X$ has the inverse-chi-square distribution with $\nu$ 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

$f(x;\nu )={\frac {2^{-\nu /2}}{\Gamma (\nu /2)}}\,x^{-\nu /2-1}e^{-1/(2x)}$ The second definition yields a density function

$f(x;\nu )={\frac {(\nu /2)^{\nu /2}}{\Gamma (\nu /2)}}x^{-\nu /2-1}e^{-\nu /(2x)}$ In both cases, $x>0$ and $\nu$ 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 $\sigma ^{2}=1/\nu$ and for the second definition $\sigma ^{2}=1$ .

## Related distributions

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