Inverse-chi-square distribution

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 =Gamma!left(frac{ u}{2},frac{1}{2x} ight)igg/, Gamma!left(frac{ u}{2} ight)!
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 eta = frac{1}{2}

ee also

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


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