Inverse-chi-square distribution

In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose multiplicative inverse (reciprocal) has a chi-square distribution. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-square distribution. That is, if $$X$$ has the chi-square distribution with $$\nu$$ degrees of freedom, then according to the first definition, $$1/X$$ has the inverse-chi-square distribution with $$\nu$$ degrees of freedom; while according to the second definition, $$\nu/X$$ has the inverse-chi-square distribution with $$\nu$$ degrees of freedom.

This distribution arises in Bayesian statistics.

It is a continuous distribution with a probability density function. The first definition yields a density function



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

The second definition yields a density function



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

In both cases, $$x>0$$ and $$\nu$$ is the degrees of freedom parameter. This article will deal with the first definition only. Both definitions are special cases of the scale-inverse-chi-square distribution. For the first definition $$\sigma^2=1/\nu$$ and for the second definition $$\sigma^2=1$$.

Related distributions

 * chi-square: If $$X \sim \chi^2(\nu)$$ and $$Y = \frac{1}{X}$$ then $$Y ~ \sim \mbox{Inv-}\chi^2(\nu)$$.
 * Inverse gamma with $$\alpha = \frac{\nu}{2}$$ and $$\beta = \frac{1}{2}$$