Inverse-chi-squared distribution
In probability and statistics, the inverse-chi-squared 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 is the probability distribution of a random variable whose multiplicative inverse has a chi-squared distribution.If follows a chi-squared distribution with degrees of [freedom (statistics)|degrees of freedom] then follows the inverse chi-squared distribution with degrees of freedom.
The probability density function of the inverse chi-squared distribution is given by
In the above and is the degrees of freedom parameter. Further, is the gamma function.
The inverse chi-squared distribution is a special case of the inverse-gamma distribution.
with shape parameter and scale parameter.
Related distributions
- chi-squared: If and, then
- scaled-inverse chi-squared: If, then
- Inverse gamma with and
- Inverse chi-squared distribution is a special case of type 5 Pearson distribution