Inverse-gamma distribution

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Inverse-gamma
Probability density function
Cumulative distribution function
Parameters α > 0 shape (real)
β > 0 scale (real)
Support x\in(0;\infty)\!
Probability density function (pdf) \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)
Cumulative distribution function (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
Excess kurtosis \frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\! for α > 4
Entropy \alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\psi(\alpha)
Moment-generating function (mgf) \frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)
Characteristic function \frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)

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.

Contents

[edit] Characterization

[edit] 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 α and scale parameter β.

[edit] 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.

[edit] Related distributions

[edit] 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 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)

[edit] See also

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Probability distributions
 
Continuous univariate supported on a semi-infinite interval
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