Inverse-chi-square distribution

Probability distribution
name =Inverse-chi-square
type =density
pdf_

cdf_

parameters =$u\; >\; 0!$
support =$x\; in\; (0,\; infty)!$
pdf =$frac\{2^\{-\; u/2${Gamma( u/2)},x^{- u/2-1} e^{-1/(2 x)}!
cdf =
mean =$frac\{1\}\{\; u-2\}!$ for $u\; >2!$
median =
mode =$frac\{1\}\{\; u+2\}!$
variance =$frac\{2\}\{(\; u-2)^2\; (\; u-4)\}!$ for $u\; >4!$
skewness =$frac\{4\}\{\; u-6\}sqrt\{2(\; u-4)\}!$ for $u\; >6!$
kurtosis =$frac\{12(5\; u-22)\}\{(\; u-6)(\; u-8)\}!$ for $u\; >8!$
entropy =$frac\{\; u\}\{2\}!+!ln!left(frac\{1\}\{2\}Gamma!left(frac\{\; u\}\{2\}\; ight)\; ight)$$!-!left(1!+!frac\{\; u\}\{2\}\; ight)psi!left(frac\{\; u\}\{2\}\; ight)$
mgf =$frac\{2\}\{Gamma(frac\{\; u\}\{2\})\}left(frac\{-t\}\{2i\}\; ight)^\{!!frac\{\; u\}\{4K\_\{frac\{\; u\}\{2!left(sqrt\{-2t\}\; ight)$
char =$frac\{2\}\{Gamma(frac\{\; u\}\{2\})\}left(frac\{-it\}\{2\}\; ight)^\{!!frac\{\; u\}\{4K\_\{frac\{\; u\}\{2!left(sqrt\{-2it\}\; ight)$
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 $u$ degrees of freedom, then according to the first definition, $1/X$ has the inverse-chi-square distribution with $u$ degrees of freedom; while according to the second definition, $u/X$ has the inverse-chi-square distribution with $u$ 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

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

The second definition yields a density function

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

In both cases, $x>0$ and $u$ 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 $sigma^2=1/\; u$ and for the second definition $sigma^2=1$.

Related distributions

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

ee also

*Scaled inverse chi-square distribution
*Chi-square distribution
*Bayesian probability