Inverse-chi-squared distribution

In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It is used in Bayesian inference as conjugate prior for the variance of the normal distribution.

Definition

The inverse chi-squared distribution (or inverted-chi-square distribution ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution.

If $X$ follows a chi-squared distribution with $\nu$ degrees of freedom then $1/X$ follows the inverse chi-squared distribution with $\nu$ degrees of freedom.

The probability density function of the inverse chi-squared distribution is given by

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

In the above $x>0$ and $\nu$ is the degrees of freedom parameter. Further, $\mathrm{\Gamma }$ is the gamma function.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter $\alpha =\frac{\nu }{2}$ and scale parameter $\beta =\frac{1}{2}$.

Related distributions

• chi-squared: If $X\sim {\chi }^2(\nu )$ and $Y=\frac{1}{X}$, then $Y\sim \text{Inv-}{\chi }^2(\nu )$
• scaled-inverse chi-squared: If $X\sim \text{Scale-inv-}{\chi }^2(\nu ,1/\nu )$, then $X\sim \text{inv-}{\chi }^2(\nu )$
• Inverse gamma with $\alpha =\frac{\nu }{2}$ and $\beta =\frac{1}{2}$
• Inverse chi-squared distribution is a special case of type 5 Pearson distribution

See also

• Scaled-inverse-chi-squared distribution
• Inverse-Wishart distribution

External links

• InvChisquare in geoR package for the R Language.