Inverse-gamma distribution

Inverse-gamma
	Probability density function;
	Cumulative distribution function;
parameters:	α > 0 shape (real); β > 0 scale (real)
support:
pdf:
cdf:
mean:	for α > 1
median:
mode:
variance:	for α > 2
skewness:	for α > 3
kurtosis:	for α > 4
entropy:
mgf:
cf:

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.

[edit] Characterization

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

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

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.

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
If $X \sim \textrm{Inv-Gamma}(1/2,c/2)$ , then $X \sim \textrm{Levy}(c)$ is a Lévy distribution; also $X \sim \textrm{Stable}(1/2,1,c,0) \,$ is a stable distribution with $α = 1 / 2$ and $β = 1$
A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart 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 $α$ ; $θ - 1$ with $β$ ; and $y$ with $x$ results in the inverse-gamma pdf shown above

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

Probability density function
Cumulative distribution function
parameters:	$α > 0$ shape (real) $β > 0$ scale (real)
support:	$x\in(0;\infty)\!$
pdf:	$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$
cdf:	$\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!$
mean:	$\frac{\beta}{\alpha-1}\!$ for $α > 1$
median:
mode:	$\frac{\beta}{\alpha+1}\!$
variance:	$\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!$ for $α > 2$
skewness:	$\frac{4\sqrt{\alpha-2}}{\alpha-3}\!$ for $α > 3$
kurtosis:	$\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!$ for $α > 4$
entropy:	$\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\psi(\alpha)$
mgf:	$\frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)$
cf:	$\frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)$