Let $X$ and $Y$ be some random variables. How do I find their joint distribution?

If I would have the joint distribution, I would find $X$ and $Y$ by integrating over the other variable. Also, if $X$ and $Y$ were independent, the joint distribution would be the product of the distributions of $X$ and $Y$. However, do I find the joint distribution in the general case, when $X$ and $Y$ are not necessarily independent?

  • $\begingroup$ If they're not independent you should know the conditional distribution(s) and then you can just simply combine one conditional distribution with the marginal distribution of the other random variable, then you get the joint distribution. $\endgroup$
    – tho_mi
    Dec 8, 2015 at 12:41
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    $\begingroup$ The marginals alone do not determine the joint, so you can't. You can, maybe, put some bounds, like the Frechet-Hoeffding bounds (read up on copulas). $\endgroup$ Dec 8, 2015 at 12:41
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    $\begingroup$ Closely related is Is it possible to have a pair of Gaussian random variables for which the joint distribution is not Gaussian? This should explain to you why your request to find the joint distribution from the marginals is impossible. $\endgroup$
    – Silverfish
    Dec 8, 2015 at 12:54