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{{Probability distribution|
== ヴィトン 出張 長財布 ==
  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.
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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]].
== 人気ショルダー財布 ヴィトン ==


==Characterization==
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===Probability density function===
The inverse gamma distribution's [[probability density function]] is defined over the [[support (mathematics)|support]] <math>x > 0</math>


:<math>
== ヴィトン レザー 財布 ==
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>.
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===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]]

Latest revision as of 08:06, 12 December 2014

ヴィトン 出張 長財布

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人気ショルダー財布 ヴィトン

proper over the G lines: new york municipality's beach If this has been time as you're stopped at the lengthy City and this rests equally distance with the 21st St. r educate stay in a queen you have for a delight. The once stockroom loaded marketplace will now be abode just to brilliant recent podiums along with also the striking Gantry Plaza area school yard, regrettably a bunch of tremendous cafes and as a consequence drinks agents,

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ヴィトン レザー 財布

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