Inverse-chi-squared distribution

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 arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown 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



f(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)} $$

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 $$\alpha = \frac{\nu}{2}$$ and scale parameter $$\beta = \frac{1}{2}$$.

Related distributions

 * chi-squared: If $$X \thicksim \chi^2(\nu)$$ and $$Y = \frac{1}{X}$$, then $$Y \thicksim \text{Inv-}\chi^2(\nu)$$
 * scaled-inverse chi-squared: If $$X \thicksim \text{Scale-inv-}\chi^2(\nu, 1/\nu) \, $$, then $$X \thicksim \text{inv-}\chi^2(\nu)$$
 * Inverse gamma with $$\alpha = \frac{\nu}{2}$$ and $$\beta = \frac{1}{2}$$
 * Inverse chi-squared distribution is a special case of type 5 Pearson distribution