Inverse-chi-squared distribution

{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!</math>|

cdf =<math>\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)

\bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!</math>|

mean =<math>\frac{1}{\nu-2}\!</math> for <math>\nu >2\!</math>|

median = <math>\approx \dfrac{1}{\nu\bigg(1-\dfrac{2}{9\nu}\bigg)^3}</math>|

mode =<math>\frac{1}{\nu+2}\!</math>|

variance =<math>\frac{2}{(\nu-2)^2 (\nu-4)}\!</math> for <math>\nu >4\!</math>|

skewness =<math>\frac{4}{\nu-6}\sqrt{2(\nu-4)}\!</math> for <math>\nu >6\!</math>|

kurtosis =<math>\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!</math> for <math>\nu >8\!</math>|

entropy =<math>\frac{\nu}{2}

\!+\!\ln\!\left(\frac{\nu}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)</math>

<math>\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)</math>|

mgf =<math>\frac{2}{\Gamma(\frac{\nu}{2})}

\left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4

K_{\frac{\nu}{2\!\left(\sqrt{-2t}\right)</math>; does not exist as real valued function|

char =<math>\frac{2}{\Gamma(\frac{\nu}{2})}

\left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4

K_{\frac{\nu}{2\!\left(\sqrt{-2it}\right)</math>|

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