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 that represents the reciprocal of 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

[edit] Characterization

[edit] Probability density function

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

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

with shape parameter $α$ and scale parameter $β$ .

[edit] Cumulative distribution function

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

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

[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

	Probability distributions [ view • talk • edit ]
	Univariate	Multivariate
Discrete:	Benford • Bernoulli • binomial • Boltzmann • categorical • compound Poisson • degenerate • Gauss-Kuzmin • geometric • hypergeometric • logarithmic • negative binomial • parabolic fractal • Poisson • Rademacher • Skellam • uniform • Yule-Simon • zeta • Zipf • Zipf-Mandelbrot	Ewens • multinomial • multivariate Polya
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 (scaled inverse chi-square)• inverse Gaussian • inverse gamma (scaled inverse gamma) • Kumaraswamy • Landau • Laplace • Lévy • Lévy skew alpha-stable • logistic • log-normal • Maxwell-Boltzmann • Maxwell speed • normal (Gaussian) • normal inverse Gaussian • Pareto • Pearson • polar • raised cosine • Rayleigh • relativistic Breit-Wigner • Rice • shifted Gompertz • Student's t • triangular • type-1 Gumbel • type-2 Gumbel • uniform • Variance-Gamma • Voigt • von Mises • Weibull • Wigner semicircle • Wilks' lambda	Dirichlet • inverse-Wishart • Kent • matrix normal • multivariate normal • multivariate Student • 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