# Inverse-chi-square distribution

Inverse-chi-square distribution

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

cdf_

parameters =$u > 0!$
support =$x in \left(0, infty\right)!$
pdf =$frac\left\{2^\left\{- u/2$
{Gamma( u/2)},x^{- u/2-1} e^{-1/(2 x)}!
cdf =
mean =$frac\left\{1\right\}\left\{ u-2\right\}!$ for $u >2!$
median =
mode =$frac\left\{1\right\}\left\{ u+2\right\}!$
variance =$frac\left\{2\right\}\left\{\left( u-2\right)^2 \left( u-4\right)\right\}!$ for $u >4!$
skewness =$frac\left\{4\right\}\left\{ u-6\right\}sqrt\left\{2\left( u-4\right)\right\}!$ for $u >6!$
kurtosis =$frac\left\{12\left(5 u-22\right)\right\}\left\{\left( u-6\right)\left( u-8\right)\right\}!$ for $u >8!$
entropy =$frac\left\{ u\right\}\left\{2\right\}!+!ln!left\left(frac\left\{1\right\}\left\{2\right\}Gamma!left\left(frac\left\{ u\right\}\left\{2\right\} ight\right) ight\right)$$!-!left\left(1!+!frac\left\{ u\right\}\left\{2\right\} ight\right)psi!left\left(frac\left\{ u\right\}\left\{2\right\} ight\right)$
mgf =$frac\left\{2\right\}\left\{Gamma\left(frac\left\{ u\right\}\left\{2\right\}\right)\right\}left\left(frac\left\{-t\right\}\left\{2i\right\} ight\right)^\left\{!!frac\left\{ u\right\}\left\{4K_\left\{frac\left\{ u\right\}\left\{2!left\left(sqrt\left\{-2t\right\} ight\right)$
char =$frac\left\{2\right\}\left\{Gamma\left(frac\left\{ u\right\}\left\{2\right\}\right)\right\}left\left(frac\left\{-it\right\}\left\{2\right\} ight\right)^\left\{!!frac\left\{ u\right\}\left\{4K_\left\{frac\left\{ u\right\}\left\{2!left\left(sqrt\left\{-2it\right\} ight\right)$
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 $u$ 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

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

The second definition yields a density function

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

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

ee also

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

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Scale-inverse-chi-square distribution — Probability distribution name =Scale inverse chi square type =density pdf cdf parameters = u > 0, sigma^2 > 0, support =x in (0, infty) pdf =frac{(sigma^2 u/2)^{ u/2{Gamma( u/2)} frac{expleft [ frac{ u sigma^2}{2 x} ight] }{x^{1+ u/2 cdf… …   Wikipedia

• Chi-square distribution — Probability distribution name =chi square type =density pdf cdf parameters =k > 0, degrees of freedom support =x in [0; +infty), pdf =frac{(1/2)^{k/2{Gamma(k/2)} x^{k/2 1} e^{ x/2}, cdf =frac{gamma(k/2,x/2)}{Gamma(k/2)}, mean =k, median… …   Wikipedia

• Chi-squared distribution — This article is about the mathematics of the chi squared distribution. For its uses in statistics, see chi squared test. For the music group, see Chi2 (band). Probability density function Cumulative distribution function …   Wikipedia

• Noncentral chi-squared distribution — Noncentral chi squared Probability density function Cumulative distribution function parameters …   Wikipedia

• Inverse-gamma distribution — Probability distribution name =Inverse gamma type =density pdf cdf parameters =alpha>0 shape (real) eta>0 scale (real) support =xin(0;infty)! pdf =frac{eta^alpha}{Gamma(alpha)} x^{ alpha 1} exp left(frac{ eta}{x} ight) cdf… …   Wikipedia

• Pearson distribution — The Pearson distribution is a family of continuous probability distributions. It was first published by Karl Pearson in 1895 and subsequently extended by him in 1901 and 1916 in a series of articles on biostatistics. History The Pearson system… …   Wikipedia

• Lévy distribution — Probability distribution name =Lévy (unshifted) type =density pdf cdf parameters =c > 0, support =x in [0, infty) pdf =sqrt{frac{c}{2pi frac{e^{ c/2x{x^{3/2 cdf = extrm{erfc}left(sqrt{c/2x} ight) mean =infinite median =c/2( extrm{erf}^{… …   Wikipedia

• Generalised hyperbolic distribution — Probability distribution name =generalised hyperbolic type =density pdf cdf parameters =mu location (real) lambda (real) alpha (real) eta asymmetry parameter (real) delta scale parameter (real) gamma = sqrt{alpha^2 eta^2} support =x in ( infty; …   Wikipedia

• Wishart distribution — Probability distribution name =Wishart type =density pdf cdf parameters = n > 0! deg. of freedom (real) mathbf{V} > 0, scale matrix ( pos. def) support =mathbf{W}! is positive definite pdf =frac{left|mathbf{W} ight|^frac{n p 1}{2… …   Wikipedia

• Student's t-distribution — Probability distribution name =Student s t type =density pdf cdf parameters = u > 0 degrees of freedom (real) support =x in ( infty; +infty)! pdf =frac{Gamma(frac{ u+1}{2})} {sqrt{ upi},Gamma(frac{ u}{2})} left(1+frac{x^2}{ u} ight)^{ (frac{… …   Wikipedia