Inverse-chi-square distribution
Encyclopedia : I : IN : INV : Inverse-chi-square distribution
\,x^ e^]| cdf =[\Gamma\left(\frac,\frac\right)/\Gamma\left(\frac\right)]|
mean =[\frac] for [\nu >2\,]| median =| mode =[\frac]| variance =[\frac] for [\nu >4\,]| skewness =[\frac\sqrt]for [\nu >6\,]| kurtosis =[\frac]for [\nu >8\,]| entropy =[\frac\!+\!\ln\left(\frac\Gamma\left(\frac\right)\right)][\!-\!\left(1\!+\!\frac\right)\psi\left(\frac\right)]|
mgf =[\frac)}\left(\frac\right)^}\!\!K_}\left(\sqrt\right)]| char =[\frac)}\left(\frac\right)^}\!\!K_}\left(\sqrt\right)]|}} In probability and statistics, the inverse-chi-square distribution is the probability distribution of a random variable whose inverse has a chi-square distribution. It is also often defined as the distribution of a random variable whose inverse divided by its degrees of freedom is a chi-square distribution. That is, if [X] has the chi-square distribution with [\nu] degrees of freedom, then according to the first definition, [1/X] has the inverse-chi-square distribution with [\nu] degrees of freedom; while according to the second definition, [\nu/X] has the inverse-chi-square distribution with [\nu] degrees of freedom.
This distribution arises in Bayesian statistics (spam filtering in particular).
It is a continuous distribution with a probability density function. The first definition yields a density function
- [ f(x; \nu)=\frac}\,x^ e^]
- [ f(x; \nu)=\frac} x^ e^]
Related distributions
- chi-square: If [X \sim \chi^2(\nu)] and [Y = \frac] then [Y ~ \sim \mbox\chi^2(\nu)].
- Inverse gamma with [\alpha = \frac] and [\beta = \frac]
See also
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