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.

1 Characterization
- 1.1 Probability density function
- 1.2 Cumulative distribution function
2 Related distributions
3 Derivation from Gamma distribution
4 See also

Characterization

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 $β$ .

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 $α$ ; $θ - 1$ with $β$ ; and $y$ with $x$ results in the inverse-gamma pdf shown above

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

Beta prime · Burr · chi-square · Coxian · Erlang · exponential · F · Fermi-Dirac · folded normal · Fréchet · Gamma · generalized extreme value · generalized inverse Gaussian · half-logistic · half-normal · Hotelling's T-square · hyper-exponential · hypoexponential · inverse chi-square (scale inverse chi-square) · inverse Gaussian · inverse gamma · Lévy · log-normal · log-logistic · Maxwell-Boltzmann · Maxwell speed · Nakagami · noncentral chi-square · Pareto · phase-type · Rayleigh · relativistic Breit–Wigner · Rice · Rosin–Rammler · shifted Gompertz · truncated normal · type-2 Gumbel · Weibull · Wilks' lambda

Continuous univariate supported on the whole real line (-∞,∞)

Cauchy · extreme value · exponential power · Fisher's z · Fisher-Tippett · generalized hyperbolic · hyperbolic secant · Landau · Laplace · Lévy skew alpha-stable · logistic · normal (Gaussian) · normal inverse Gaussian · skew normal · Student's t · type-1 Gumbel · Variance-Gamma · Voigt

Multivariate (joint)

Discrete: Ewens · Beta-binomial · multinomial · multivariate Polya Continuous: Dirichlet · Generalized Dirichlet · multivariate normal · multivariate Student · normal-scaled inverse gamma · normal-gamma Matrix-valued: inverse-Wishart · matrix normal · Wishart

Directional, degenerate, and singular

Directional: Kent · von Mises · von Mises–Fisher Degenerate: discrete degenerate · Dirac delta function Singular: Cantor

Families

exponential · natural exponential · location-scale · maximum entropy · Pearson · Tweedie

Wikipedia content modification information:

This page was last modified on 21 October 2008, at 18:21.

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