Inverse-gamma distribution

Probability distribution
name =Inverse-gamma
type =density
pdf_

cdf_

parameters =$alpha>0$ shape (real)
scale (real)
support =$xin(0;infty)!$
pdf =
cdf =
mean = for $alpha\; >\; 1$
median =
mode =
variance = for $alpha\; >\; 2$
skewness =$frac\{4sqrt\{alpha-2${alpha-3}! for $alpha\; >\; 3$
kurtosis =$frac\{30,alpha-66\}\{(alpha-3)(alpha-4)\}!$ for $alpha\; >\; 4$
entropy =
mgf =
char =

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.

Characterization

Probability density function

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

:

with shape parameter $alpha$ and scale parameter .

Cumulative distribution function

The cumulative distribution function is the regularized gamma function

:

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

Related distributions

* If and $alpha\; =\; frac\{\; u\}\{2\}$ and then $X\; sim\; mbox\{Inv-chi-square\}(\; u),$ is an inverse-chi-square distribution
* If $X\; sim\; mbox\{Inv-Gamma\}(k,\; heta),$ , then $1/X\; sim\; mbox\{Gamma\}(k,\; heta^\{-1\}),$ 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

:$f(x)\; =\; x^\{k-1\}\; frac\{e^\{-x/\; heta\{\; heta^k\; ,\; Gamma(k)\}$

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

:$f\_Y(y)\; =\; f\_X\; left(\; g^\{-1\}(y)\; ight)\; left|\; frac\{d\}\{dy\}\; g^\{-1\}(y)\; ight$
::$=frac\{1\}\{\; heta^k\; Gamma(k)\}left(\; frac\{1\}\{y\}\; ight)^\{k-1\}exp\; left(\; frac\{-1\}\{\; heta\; y\}\; ight)frac\{1\}\{y^2\}$::$=frac\{1\}\{\; heta^k\; Gamma(k)\}left(\; frac\{1\}\{y\}\; ight)^\{k+1\}exp\; left(\; frac\{-1\}\{\; heta\; y\}\; ight)$::$=frac\{1\}\{\; heta^k\; Gamma(k)\}y^\{-k-1\}exp\; left(\; frac\{-1\}\{\; heta\; y\}\; ight).$

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

:

ee also

*gamma distribution
*inverse-chi-square distribution
*normal distribution