Английская Википедия:Inverse-chi-squared distribution
Шаблон:Short description Шаблон:Probability distribution{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}\!</math>|
cdf =<math>\Gamma\!\left(\frac{\nu}{2},\frac{1}{2x}\right)
\bigg/\, \Gamma\!\left(\frac{\nu}{2}\right)\!</math>|
mean =<math>\frac{1}{\nu-2}\!</math> for <math>\nu >2\!</math>|
median = <math>\approx \dfrac{1}{\nu\bigg(1-\dfrac{2}{9\nu}\bigg)^3}</math>|
mode =<math>\frac{1}{\nu+2}\!</math>|
variance =<math>\frac{2}{(\nu-2)^2 (\nu-4)}\!</math> for <math>\nu >4\!</math>|
skewness =<math>\frac{4}{\nu-6}\sqrt{2(\nu-4)}\!</math> for <math>\nu >6\!</math>|
kurtosis =<math>\frac{12(5\nu-22)}{(\nu-6)(\nu-8)}\!</math> for <math>\nu >8\!</math>|
entropy =<math>\frac{\nu}{2}
\!+\!\ln\!\left(\frac{\nu}{2}\Gamma\!\left(\frac{\nu}{2}\right)\right)</math> <math>\!-\!\left(1\!+\!\frac{\nu}{2}\right)\psi\!\left(\frac{\nu}{2}\right)</math>|
mgf =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
\left(\frac{-t}{2i}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2t}\right)</math>; does not exist as real valued function|
char =<math>\frac{2}{\Gamma(\frac{\nu}{2})}
\left(\frac{-it}{2}\right)^{\!\!\frac{\nu}{4}} K_{\frac{\nu}{2}}\!\left(\sqrt{-2it}\right)</math>| }}
In probability and statistics, the inverse-chi-squared distribution (or inverted-chi-square distribution[1]) is a continuous probability distribution of a positive-valued random variable. It is closely related to the chi-squared distribution. It arises in Bayesian inference, where it can be used as the prior and posterior distribution for an unknown variance of the normal distribution.
Definition
The inverse-chi-squared distribution (or inverted-chi-square distribution[1] ) 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, 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. Information associated with the first definition is depicted on the right side of the page.
The first definition yields a probability density function given by
- <math>
f_1(x; \nu) = \frac{2^{-\nu/2}}{\Gamma(\nu/2)}\,x^{-\nu/2-1} e^{-1/(2 x)}, </math>
while the second definition yields the density function
- <math>
f_2(x; \nu) = \frac{(\nu/2)^{\nu/2}}{\Gamma(\nu/2)} x^{-\nu/2-1} e^{-\nu/(2 x)} . </math>
In both cases, <math>x>0</math> and <math>\nu</math> is the 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>.
Related distributions
- 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>
- 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>
- Inverse gamma with <math>\alpha = \frac{\nu}{2}</math> and <math>\beta = \frac{1}{2}</math>
- Inverse chi-squared distribution is a special case of type 5 Pearson distribution
See also
References
External links
- InvChisquare in geoR package for the R Language.
- ↑ 1,0 1,1 Bernardo, J.M.; Smith, A.F.M. (1993) Bayesian Theory, Wiley (pages 119, 431) Шаблон:ISBN
