Inverse-gamma distribution

From Wikipedia, the free encyclopedia

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)
mgf \frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)
Char. func. \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 that represents the reciprocal of the gamma distribution.

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

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

with shape parameter α and scale parameter β.

The cumulative distribution function is

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] Gamma distribution derivation

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

Image:Bvn-small.png Probability distributions [ view • talk • edit ]
Univariate Multivariate
Discrete: Bernoulli • binomial • Boltzmann • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot Ewens • multinomial
Continuous: Beta • Beta prime • Cauchy • chi-square • Dirac delta function • Erlang • exponential • exponential power • F • fading • Fisher's z • Fisher-Tippett • Gamma • generalized extreme value • generalized hyperbolic • generalized inverse Gaussian • Half-Logistic • Hotelling's T-square • hyperbolic secant • hyper-exponential • hypoexponential • inverse chi-square • inverse Gaussian • inverse gamma • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda Dirichlet • Kent • matrix normal • multivariate normal • von Mises-Fisher • Wigner quasi • Wishart
Miscellaneous: Cantor • conditional • exponential family • infinitely divisible • location-scale family • marginal • maximum entropy • phase-type • posterior • prior • quasi • sampling • singular
Retrieved from "http://en.wikipedia.org../../../i/n/v/Inverse-gamma_distribution.html"
In other languages