Symbolic integration: Difference between revisions
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{{Probability distribution| | |||
name =Inverse-gamma| | |||
type =density| | |||
pdf_image =[[Image:Inverse gamma pdf.png|325px]]| | |||
cdf_image =[[Image:Inverse gamma cdf.png|325px]]| | |||
parameters =<math>\alpha>0</math> [[shape parameter|shape]] ([[real number|real]])<br /><math>\beta>0</math> [[scale parameter|scale]] (real)| | |||
support =<math>x\in(0;\infty)\!</math>| | |||
pdf =<math>\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)</math>| | |||
cdf =<math>\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!</math>| | |||
mean =<math>\frac{\beta}{\alpha-1}\!</math> for <math>\alpha > 1</math>| | |||
median =| | |||
mode =<math>\frac{\beta}{\alpha+1}\!</math>| | |||
variance =<math>\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!</math> for <math>\alpha > 2</math>| | |||
skewness =<math>\frac{4\sqrt{\alpha-2}}{\alpha-3}\!</math> for <math>\alpha > 3</math>| | |||
kurtosis =<math>\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!</math> for <math>\alpha > 4</math>| | |||
entropy =<math>\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)</math>| | |||
mgf =<math>\frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)</math>; does not exist as [[real number|real valued]] function| | |||
char =<math>\frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)</math>| | |||
}} | |||
In [[probability theory]] and [[statistics]], the '''inverse gamma distribution''' is a two-parameter family of continuous [[probability distribution]]s on the positive [[real line]], which is the distribution of the [[multiplicative inverse|reciprocal]] of a variable distributed according to the [[gamma distribution]]. Perhaps the chief use of the inverse gamma distribution is in [[Bayesian statistics]], where the distribution arises as the marginal posterior distribution for the unknown [[variance]] of a [[normal distribution]] if an [[uninformative prior]] is used; and as an analytically tractable [[conjugate prior]] if an informative prior is required. | |||
However, it is common among Bayesians to consider an alternative [[parametrization]] of the [[normal distribution]] in terms of the [[precision (statistics)|precision]], defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a [[scaled inverse chi-squared distribution]]. | |||
==Characterization== | |||
===Probability density function=== | |||
The inverse gamma distribution's [[probability density function]] is defined over the [[support (mathematics)|support]] <math>x > 0</math> | |||
:<math> | |||
f(x; \alpha, \beta) | |||
= \frac{\beta^\alpha}{\Gamma(\alpha)} | |||
x^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right) | |||
</math> | |||
with [[shape parameter]] <math>\alpha</math> and <!-- inverse scale? --> [[scale parameter]] <math>\beta</math>. | |||
===Cumulative distribution function=== | |||
The [[cumulative distribution function]] is the [[Incomplete gamma function#Regularized Gamma functions and Poisson random variables|regularized gamma function]] | |||
:<math>F(x; \alpha, \beta) = \frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(\alpha)} = Q\left(\alpha, \frac{\beta}{x}\right)\!</math> | |||
where the numerator is the upper [[incomplete gamma function]] and the denominator is the [[gamma function]]. Many math packages allow you to compute Q, the regularized gamma function, directly. | |||
===Characteristic function=== | |||
<math>K_{\alpha}(\cdot)</math> in the expression of the [[characteristic function (probability theory)|characteristic function]] is the modified [[Bessel function]] of II kind. | |||
==Properties== | |||
For <math>\alpha>0 </math> and <math>\beta>0</math> | |||
: <math>\mathbb{E}[\ln(X)] = \ln(\beta) - \psi(\alpha).\, </math> | |||
: <math>\mathbb{E}[X^{-1}] = \frac{\alpha}{\beta}.\, </math> | |||
where <math>\psi(\alpha) </math> is the [[digamma function]]. | |||
==Related distributions== | |||
* If <math>X \sim \mbox{Inv-Gamma}(\alpha, \beta)</math> then <math> k X \sim \mbox{Inv-Gamma}(\alpha, k \beta) \,</math> | |||
* If <math>X \sim \mbox{Inv-Gamma}(\alpha, \tfrac{1}{2})</math> then <math>X \sim \mbox{Inv-}\chi^2(2 \alpha)\,</math> ([[inverse-chi-squared distribution]]) | |||
* If <math>X \sim \mbox{Inv-Gamma}(\tfrac{\alpha}{2}, \tfrac{1}{2})</math> then <math>X \sim \mbox{Scaled Inv-}\chi^2(\alpha,\tfrac{1}{\alpha})\,</math> ([[scaled-inverse-chi-squared distribution]]) | |||
* If <math>X \sim \textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})</math> then <math>X \sim \textrm{Levy}(0,c)\,</math> ([[Lévy distribution]]) | |||
* If <math>X \sim \mbox{Gamma}(k, \theta)\,</math> ([[Gamma distribution]]) then <math>\tfrac{1}{X} \sim \mbox{Inv-Gamma}(k, \theta^{-1})\,</math> (see derivation in the next paragraph for details) | |||
* If <math>X \sim \mbox{Gamma}(\alpha, \beta^{-1})\,</math> ([[Gamma distribution]]) then <math>\tfrac{1}{X} \sim \mbox{Inv-Gamma}(\alpha, \beta)\,</math> | |||
* Inverse gamma distribution is a special case of type 5 [[Pearson distribution]] | |||
* A [[multivariate]] generalization of the inverse-gamma distribution is the [[inverse-Wishart distribution]]. | |||
* For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001) | |||
==Derivation from Gamma distribution== | |||
The pdf of the [[gamma distribution]] is | |||
:<math> f(x) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)}</math> | |||
and define the transformation <math>Y = g(X) = \frac{1}{X}</math> then the resulting transformation is | |||
:<math> | |||
f_Y(y) = f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right| | |||
</math> | |||
::<math> | |||
= | |||
\frac{1}{\theta^k \Gamma(k)} | |||
\left( | |||
\frac{1}{y} | |||
\right)^{k-1} | |||
\exp | |||
\left( | |||
\frac{-1}{\theta y} | |||
\right) | |||
\frac{1}{y^2} | |||
</math> | |||
::<math> | |||
= | |||
\frac{1}{\theta^k \Gamma(k)} | |||
\left( | |||
\frac{1}{y} | |||
\right)^{k+1} | |||
\exp | |||
\left( | |||
\frac{-1}{\theta y} | |||
\right) | |||
</math> | |||
::<math> | |||
= | |||
\frac{1}{\theta^k \Gamma(k)} | |||
y^{-k-1} | |||
\exp | |||
\left( | |||
\frac{-1}{\theta y} | |||
\right). | |||
</math> | |||
Replacing <math>k</math> with <math>\alpha</math>; <math>\theta^{-1}</math> with <math>\beta</math>; and <math>y</math> with <math>x</math> results in the inverse-gamma pdf shown above | |||
:<math> | |||
f(x) | |||
= | |||
\frac{\beta^\alpha}{\Gamma(\alpha)} | |||
x^{-\alpha-1} | |||
\exp | |||
\left( | |||
\frac{-\beta}{x} | |||
\right). | |||
</math> | |||
==Occurrence== | |||
* The first [[hitting time]] of a standard [[Wiener process]] has an inverse-gamma distribution with parameters <math>\alpha = \frac{1}{2}</math> and <math>\beta = \frac{x^2}{2}</math> where <math>x</math> is the value to hit. | |||
==See also== | |||
*[[gamma distribution]] | |||
*[[inverse-chi-squared distribution]] | |||
*[[normal distribution]] | |||
==References== | |||
* V. Witkovsky (2001) ''Computing the distribution of a linear combination of inverted gamma variables'', '''Kybernetika''' 37(1), 79-90 | |||
{{ProbDistributions|continuous-semi-infinite}} | |||
{{DEFAULTSORT:Inverse-Gamma Distribution}} | |||
[[Category:Continuous distributions]] | |||
[[Category:Conjugate prior distributions]] | |||
[[Category:Probability distributions with non-finite variance]] | |||
[[Category:Exponential family distributions]] | |||
[[Category:Probability distributions]] |
Revision as of 08:23, 28 July 2013
Template:Probability distribution
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. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution if an uninformative prior is used; and as an analytically tractable conjugate prior if an informative prior is required.
However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution.
Characterization
Probability density function
The inverse gamma distribution's probability density function is defined over the support
with shape parameter 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. Many math packages allow you to compute Q, the regularized gamma function, directly.
Characteristic function
in the expression of the characteristic function is the modified Bessel function of II kind.
Properties
where is the digamma function.
Related distributions
- If then
- If then (inverse-chi-squared distribution)
- If then (scaled-inverse-chi-squared distribution)
- If then (Lévy distribution)
- If (Gamma distribution) then (see derivation in the next paragraph for details)
- If (Gamma distribution) then
- Inverse gamma distribution is a special case of type 5 Pearson distribution
- A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.
- For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)
Derivation from Gamma distribution
The pdf of the gamma distribution is
and define the transformation then the resulting transformation is
Replacing with ; with ; and with results in the inverse-gamma pdf shown above
Occurrence
- The first hitting time of a standard Wiener process has an inverse-gamma distribution with parameters and where is the value to hit.
See also
References
- V. Witkovsky (2001) Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37(1), 79-90
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