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Let the random vector $(X_1,\ldots,X_n)$ follow the multivariate uniform distribution. Can I claim that the random variables $X_1,\ldots,X_n$ must be independent uniform random variables? In other words, does there exist some random variables whose joint probability distribution is the uniform distribution but some of whose marginal distributions are not uniform?

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  • $\begingroup$ The joint distribution completely determines the marginal distributions, so yes. (Or no to your last question.) $\endgroup$ – Rahul Oct 28 '16 at 1:40
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Depends on what qualifies as a multivariate uniform distribution. If you require the joint distribution be uniform on a $n$-dimensional rectangle (a product of intervals), then yes, the marginals are independent and individually uniform. Otherwise, the marginals are not mutually independent, and some marginals need not be uniform. For example consider a 2-d uniform distribution on a non-rectangular parallelogram, or 2-d uniform on a rotated square whose sides are not parallel to the axes, or a 2-d uniform distribution on a circle....

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