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For continuoues variables, how is a coupling of two distributions different from their joint distribution? Are they the same concepts?

Update: Coupling is the same as defining a joint distribution on the Cartesian product space of the supports of original random variables, in a way that marginals of the defined joint distribution are equal to the original random variables. What am I missing?

user25004
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    Coupling of distributions has no relation to joint distributions. Coupling refers to taking random variables defined on different prob. spaces on putting equivalent variables (same distribution) on a single prob. space. https://en.wikipedia.org/wiki/Coupling_(probability) – herb steinberg Mar 25 '19 at 21:39
  • Even your explanation still seems like defining a joint distribution on the product space of the supports of original random variables, in a way that marginals of the defined joint distribution are equal to the original random variables. What am I missing? – user25004 Mar 25 '19 at 23:08
  • My understanding is that coupling means defining random variables on the same probability space with the same distribution as those on separate spaces. This is my reading of the wikipedia reference. I personally am not familiar with the term "coupling distributions".. – herb steinberg Mar 26 '19 at 02:50
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    @herbsteinberg's comment is not correct. Copulas are fundamentally about constructing joint distributions from marginals, and your edit is not missing anything except possibly the name of the theorem that proves this, Sklar's theorem. – hrrrrrr5602 May 27 '24 at 22:22

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