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Does $X \sim X'$ and $Y \sim Y'$ implies that $(X,Y) \sim (X',Y')$?

By $X \sim X'$, I mean that the random variables $X$ and $X'$ are distributed equally.

I've been thinking whether the above statement holds in general for arbitrary random variables.

The statement would be useful in doing some proofs in computer science, but although the statement seems intuitively, I don't know how to prove it rigorously.

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    No, it does not. The marginal distribution does not tell you the full information about the joint distribution. – 2017-02-13

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Let $X \sim N(0,1)$. Note that $X \sim X$ and $X \sim -X$. However $(X,X)$ does not have the same distribution as $(X,-X)$.

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    So, I guess the claim is also false, if we only consider discrete random variables (finite set)? – 2017-02-13
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    Yes, the counterexample only requires for $X$ to have a distribution that is symmetric around $0$. It can be discrete. – 2017-02-13