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I am trying to understand the concept of a joint probability distribution. According to the definition of a (say) bivariate probability distribution, the two random variables $X$ and $Y$ should be defined on the same probability space. http://en.wikipedia.org/wiki/Joint_probability_distribution

Are there such things as joint probability distributions over different sample spaces or different probability space?

usual me
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Of course you can. The joint probability has Cartesian product of each own's sample space as joint sample space and product $\sigma-$algebra as joint events space. The joint probability, obviously, is the product of probabilities of $X$ and $Y$.

Shuchang
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    The construction you describe assumes one defines new random variables $X'$ and $Y'$ on a new probability space $(\Omega',\mathcal F',P')$ such that $X'$ coincides with $X$ in distribution and $Y'$ coincides with $Y$ in distribution (by the way the way you describe is not the only one). Thus, the answer to "Are there such things as joint probability distributions over different probability spaces?" is rather: Of course there is no such thing. – Did Oct 09 '13 at 08:43
  • Well, what is your comment on [this] (http://math.stackexchange.com/questions/22007/how-to-understand-joint-distribution-of-a-random-vector) – Shuchang Oct 09 '13 at 08:51
  • Sorry but I fail to see why I should have one (comment) and how this is related to my comment above. Please explain. – Did Oct 09 '13 at 09:09
  • Sorry, I misunderstand your comment. – Shuchang Oct 09 '13 at 09:51
  • @Did, can you kindly provide a citation for your comment, ``The construction you describe assumes... in distribution''? – Michael Levy Jul 24 '19 at 15:02