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 ) is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-squared distribution. If X follows a chi-squared distribution with \nu degrees of freedom then 1/X follows the inverse chi-squared distribution with \nu degrees of freedom. The probability density function of the inverse chi-squared distribution is given by In the above x>0 and \nu is the degrees of freedom parameter. Further, \Gamma is the gamma function. The inverse chi-squared distribution is a special case of the inverse-gamma distribution. with shape parameter and scale parameter.

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