Inverse-chi-squared distribution

name =Inverse-chi-squared| type =density| pdf_image =[[Image:Inverse chi squared density.png]]| cdf_image =[[Image:Inverse chi squared distribution.png]]| parameters =\nu 0!| support =x \in (0, \infty)!| pdf =\frac{2^{-\nu/2}}{\Gamma(\nu/2)},x^{-\nu/2-1} e^{-1/(2 x)}!| 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!| 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| char =\frac{2}{\Gamma(\frac{\nu}{2})} \left(\frac{-it}{2}\right)^{!!\frac{\nu}{4}} K_{\frac{\nu}{2}}!\left(\sqrt{-2it}\right)|

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.