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.

degrees of freedom then

follows the inverse chi-squared distribution with

The probability density function of the inverse chi-squared distribution is given by In the above

is the degrees of freedom parameter.

The inverse chi-squared distribution is a special case of the inverse-gamma distribution.