Inverse-chi-squared
Probability density function
Cumulative distribution function
Parameters	${\displaystyle \nu >0}$
Support	${\displaystyle x\in (0,\mathrm{\infty })}$
PDF	${\displaystyle \frac{2^{-\nu /2}}{\mathrm{\Gamma }(\nu /2)}{x}^{-\nu /2-1}{e}^{-1/(2x)}}$
CDF	${\displaystyle \mathrm{\Gamma }\left(\frac{\nu }{2},\frac{1}{2x}\right)/\mathrm{\Gamma }\left(\frac{\nu }{2}\right)}$
Mean	${\displaystyle \frac{1}{\nu -2}}$ for${\displaystyle \nu >2}$
Median	${\displaystyle \approx {\displaystyle \frac{1}{\nu (1-{\displaystyle \frac{2}{9\nu }}{)}^3}}}$
Mode	${\displaystyle \frac{1}{\nu +2}}$
Variance	${\displaystyle \frac{2}{(\nu -2{)}^2(\nu -4)}}$ for${\displaystyle \nu >4}$
Skewness	${\displaystyle \frac{4}{\nu -6}\sqrt{2(\nu -4)}}$ for${\displaystyle \nu >6}$
Excess kurtosis	${\displaystyle \frac{12(5\nu -22)}{(\nu -6)(\nu -8)}}$ for${\displaystyle \nu >8}$
Entropy	${\displaystyle \frac{\nu }{2}+\ln \left(\frac{\nu }{2}\mathrm{\Gamma }\left(\frac{\nu }{2}\right)\right)}$ ${\displaystyle -\left(1+\frac{\nu }{2}\right)\psi \left(\frac{\nu }{2}\right)}$
MGF	${\displaystyle \frac{2}{\mathrm{\Gamma }(\frac{\nu }{2})}{\left(\frac{-t}{2i}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2t}\right)}$; does not exist asreal valued function
CF	${\displaystyle \frac{2}{\mathrm{\Gamma }(\frac{\nu }{2})}{\left(\frac{-it}{2}\right)}^{\frac{\nu }{4}}{K}_{\frac{\nu }{2}}\left(\sqrt{-2it}\right)}$

In probability and statistics, theinverse-chi-squared distribution (orinverted-chi-square distribution^[1]) is acontinuous probability distribution of a positive-valued random variable. It is closely related to thechi-squared distribution. It is used inBayesian inference asconjugate prior for thevariance of thenormal distribution.^[2]

The inverse chi-squared distribution (or inverted-chi-square distribution^[1] ) is theprobability distribution of a random variable whosemultiplicative inverse (reciprocal) has achi-squared distribution.

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

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

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

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

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

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

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

• InvChisquare in geoR package for the R Language.

• Continuous distributions
• Exponential family distributions
• Probability distributions with non-finite variance

• Articles with short description
• Short description matches Wikidata