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| {{Probability distribution|
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| name =Inverse-chi-squared|
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| type =density|
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| pdf_image =[[Image:Inverse chi squared density.png]]|
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| cdf_image =[[Image:Inverse chi squared distribution.png]]|
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| parameters =<math>\nu > 0\!</math>|
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| support =<math>x \in (0, \infty)\!</math>|
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| pdf =<math>\frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!</math>|
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| cdf =<math>\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)
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| \bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!</math>|
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| mean =<math>\frac{1}{\nu-2}\!</math> for <math>\nu >2\!</math>|
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| median =|
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| mode =<math>\frac{1}{\nu+2}\!</math>|
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| variance =<math>\frac{2}{(\nu-2)^2 (\nu-4)}\!</math> for <math>\nu >4\!</math>|
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| skewness =<math>\frac{4}{\nu-6}\sqrt{2(\nu-4)}\!</math> for <math>\nu >6\!</math>|
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| kurtosis =<math>\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!</math> for <math>\nu >8\!</math>|
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| entropy =<math>\frac{\nu}{2}
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| \!+\!\ln\!\left(\frac{1}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)</math>
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| <math>\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)</math>|
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| mgf =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
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| \left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}}
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| K_{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)</math>; does not exist as [[real number|real valued]] function|
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| char =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
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| \left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}}
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| K_{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)</math>|
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| }}
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| In [[probability and statistics]], the '''inverse-chi-squared distribution''' (or '''inverted-chi-square distribution'''<ref name=BS>Bernardo, J.M.; Smith, A.F.M. (1993) ''Bayesian Theory'' ,Wiley (pages 119, 431) ISBN 0-471-49464-X</ref>) is a [[continuous probability distribution]] of a positive-valued random variable. It is closely related to the [[chi-squared distribution]] and its specific importance is that it arises in the application of [[Bayesian inference]] to the [[normal distribution]], where it can be used as the [[prior distribution|prior]] and [[posterior distribution]] for an unknown [[variance]].
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| ==Definition==
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| The inverse-chi-squared distribution (or inverted-chi-square distribution<ref name=BS/> ) is the [[probability distribution]] of a random variable whose [[multiplicative inverse]] (reciprocal) has a [[chi-squared distribution]]. It is also often defined as the distribution of a random variable whose reciprocal divided by its degrees of freedom is a chi-squared distribution. That is, if <math>X</math> has the chi-squared distribution with <math>\nu</math> [[degrees of freedom (statistics)|degrees of freedom]], then according to the first definition, <math>1/X</math> has the inverse-chi-squared distribution with <math>\nu</math> degrees of freedom; while according to the second definition, <math>\nu/X</math> has the inverse-chi-squared distribution with <math>\nu</math> degrees of freedom. Only the first definition will usually be covered in this article.
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| The first definition yields a [[probability density function]] given by
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| :<math>
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| f_1(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)},
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| </math>
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| while the second definition yields the density function
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| :<math>
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| f_2(x; \nu) =
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| \frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)} x^{-\nu/2-1} e^{-\nu/(2 x)} .
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| </math>
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| In both cases, <math>x>0</math> and <math>\nu</math> is the [[degrees of freedom (statistics)|degrees of freedom]] parameter. Further, <math>\Gamma</math> is the [[gamma function]]. Both definitions are special cases of the [[scaled-inverse-chi-squared distribution]]. For the first definition the variance of the distribution is <math>\sigma^2=1/\nu ,</math> while for the second definition <math>\sigma^2=1</math>.
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| ==Related distributions==
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| *[[chi-squared distribution|chi-squared]]: If <math>X \thicksim \chi^2(\nu)</math> and <math>Y = \frac{1}{X}</math>, then <math>Y \thicksim \text{Inv-}\chi^2(\nu)</math>
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| *[[scaled-inverse-chi-squared distribution|scaled-inverse chi-squared]]: If <math>X \thicksim \text{Scale-inv-}\chi^2(\nu, 1/\nu) \, </math>, then <math>X \thicksim \text{inv-}\chi^2(\nu)</math>
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| *[[Inverse-gamma distribution|Inverse gamma]] with <math>\alpha = \frac{\nu}{2}</math> and <math>\beta = \frac{1}{2}</math>
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| ==See also==
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| *[[Scaled-inverse-chi-squared distribution]]
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| *[[Inverse-Wishart distribution]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://hosho.ees.hokudai.ac.jp/~kubo/Rdoc/library/geoR/html/InvChisquare.html InvChisquare] in geoR package for the R Language.
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| {{ProbDistributions|continuous-semi-infinite}}
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| [[Category:Continuous distributions]]
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| [[Category:Exponential family distributions]]
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| [[Category:Probability distributions with non-finite variance]]
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| [[Category:Probability distributions]]
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