Multivariate gamma function
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In mathematics, the multivariate Gamma distribution, Γp(·), is a generalization of the Gamma function. It is useful in multivariate statistics.
It has two equivalent definitions. One is
- <math>
\Gamma_p(a)= \int_{S\in {\mathbf S}} \exp\left( -{\rm trace}(S)\right) \left|S\right|^{a-(p+1)/2} dS </math> where S is the set of all positive-definite matrices. The other one, more useful in practice, is
- <math>
\Gamma_p(a)= \pi^{p(p-1)/4}\prod_{j=1}^p \Gamma\left[ a+(1-j)/2\right]. </math>
Thus
- <math>\Gamma_1(a)=\Gamma(a)</math>
- <math>\Gamma_2(a)=\pi^{1/2}\Gamma(a)\Gamma(a-1/2)</math>
- <math>\Gamma_3(a)=\pi^{3/2}\Gamma(a)\Gamma(a-1/2)\Gamma(a-1)</math>
and so on.eo:Vikipedio:Projekto matematiko/Multvariebla γ funkcio