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I'm interested in multivariate probability distributions with different sub-distributions. Say I've collected two metrics in a population, and we then had a two dimensional probability distribution:

X1 ~ Normal(U, V) and X2 ~ Poi(Lambda)

Notice how these two variables come from different probability distributions. I've been having trouble finding reference material for multivariate analysis where the dimensions come from different distributions.

For example: the Wischart distribution works well when the variables all come from chi-square distributions. Does anyone have any reference material for probability distributions where the sub-distributions are not the same?

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  • $\begingroup$ What is "sub-distribution"? Did you mean marginals? $\endgroup$ – Aksakal Mar 14 '16 at 15:23
  • $\begingroup$ Hmm I suppose so $\endgroup$ – user46925 Mar 14 '16 at 15:33
  • $\begingroup$ So, will your joint distribution be something like $(x_1,x_2)\sim f(u,v,\lambda)?$ $\endgroup$ – Aksakal Mar 14 '16 at 15:47
  • $\begingroup$ Yeah, that would be it! $\endgroup$ – user46925 Mar 14 '16 at 15:52
  • $\begingroup$ Is that still a Copula? They may or may not be related variables. (I don't know yet - but I do know that one follows a Poisson and another follows a Gaussian) $\endgroup$ – user46925 Mar 14 '16 at 15:52
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If by "sub-dsitributions" in your question you mean marginals, then read about copulas.

The idea's that if you know the marginal distributions, then for any joint distribution there's an object called copula, which links the marginals to the joint distribution. It's like a functional, whose inputs are marginal distributions, and the output is the joint.

It's an elegant construct, the only issue is that you usually don't know which copula is the right one to use. So, you end up using the one you like and hope that it's the one.

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  • $\begingroup$ Interesting! I would have never found that out. What a cool word too! $\endgroup$ – user46925 Mar 14 '16 at 15:29
  • $\begingroup$ Is the Wishart distribution a type of Copula? $\endgroup$ – user46925 Mar 14 '16 at 15:31
  • $\begingroup$ The distribution is not a copula. So, no Wishart is not a copula. The copula is a construct that links univariates into a joint distribution. $\endgroup$ – Aksakal Mar 14 '16 at 15:35
  • $\begingroup$ Hm, I don't quite understand: we can express the Wischart as marginal probabilties (one for each dimension) - hence I believe it could be expressed as a Copula. $\endgroup$ – user46925 Mar 14 '16 at 15:37
  • $\begingroup$ If there another term, in the case that the "sub-distributions" are somehow related? $\endgroup$ – user46925 Mar 14 '16 at 15:37

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