Inverse-chi-squared distribution

Inverse-chi-squared
	Probability density function
	Cumulative distribution function
Parameters
Support
PDF
CDF
Mean	for
Median
Mode
Variance	for
Skewness	for
Ex. kurtosis	for
Entropy
MGF	; does not exist as real valued function
CF

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 arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution.

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