ν 2 + ln ( ν 2 Γ ( ν 2 ) ) {\displaystyle {\frac {\nu }{2}}\!+\!\ln \!\left({\frac {\nu }{2}}\Gamma \!\left({\frac {\nu }{2}}\right)\right)}
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution[1]) 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.[2]
The inverse chi-squared distribution (or inverted-chi-square distribution[1] ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution.
If X {\displaystyle X} follows a chi-squared distribution with ν {\displaystyle \nu } degrees of freedom then 1 / X {\displaystyle 1/X} follows the inverse chi-squared distribution with ν {\displaystyle \nu } degrees of freedom.
The probability density function of the inverse chi-squared distribution is given by
In the above x > 0 {\displaystyle x>0} and ν {\displaystyle \nu } is the degrees of freedom parameter. Further, Γ {\displaystyle \Gamma } is the gamma function.
The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter α = ν 2 {\displaystyle \alpha ={\frac {\nu }{2}}} and scale parameter β = 1 2 {\displaystyle \beta ={\frac {1}{2}}} .
