Inverse-gamma distribution

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.

Probability density function
The inverse gamma distribution's probability density function is defined over the support $$x > 0$$



f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} (1/x)^{\alpha + 1}\exp\left(-\beta/x\right) $$

with shape parameter $$\alpha$$ and scale parameter $$\beta$$.

Cumulative distribution function
The cumulative distribution function is the regularized gamma function


 * $$F(x; \alpha, \beta) = \frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!$$

where the numerator is the upper incomplete gamma function and the denominator is the gamma function.

Related distributions

 * If $$X \sim \mbox{Inv-Gamma}(\alpha, \beta)$$ and $$\alpha = \frac{\nu}{2}$$ and $$\beta = \frac{1}{2}$$ then $$X \sim \mbox{Inv-chi-square}(\nu)\,$$ is an inverse-chi-square distribution
 * If $$X \sim \mbox{Inv-Gamma}(k, \theta)\,$$, then $$1/X \sim \mbox{Gamma}(k, \theta^{-1})\,$$ is a Gamma distribution
 * A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.

Derivation from Gamma distribution
The pdf of the gamma distribution is


 * $$ f(x) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)}$$

and define the transformation $$Y = g(X) = \frac{1}{X}$$ then the resulting transformation is



f_Y(y) = f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right| $$

= \frac{1}{\theta^k \Gamma(k)} \left( \frac{1}{y} \right)^{k-1} \exp \left( \frac{-1}{\theta y} \right) \frac{1}{y^2} $$

= \frac{1}{\theta^k \Gamma(k)} \left( \frac{1}{y} \right)^{k+1} \exp \left( \frac{-1}{\theta y} \right) $$

= \frac{1}{\theta^k \Gamma(k)} y^{-k-1} \exp \left( \frac{-1}{\theta y} \right) $$

Replacing $$k$$ with $$\alpha$$; $$\theta^{-1}$$ with $$\beta$$; and $$y$$ with $$x$$ results in the inverse-gamma pdf shown above



f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha-1} \exp \left( \frac{-\beta}{x} \right) $$