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So I know one can go from a joint density function $f(x,y)$ to marginal density functions, like $f_x(x)$ by integrating against the other variables as in $f_x(x) = \int f(x,y) dy$...but given $f_x(x)$ and $f_y(y)$ as densities for dependent random vars..how would one go about finding a joint density or distribution function?

Thanks

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    In general, you can't recover the joint density from the marginals. You need more information. In a few cases (eg joint gaussian variables, or two bernoullis) the correlation would suffice.2011-10-30

2 Answers 2

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For example, suppose the marginal densities for $X$ and $Y$ are both 1 on the interval $[0,1]$, 0 otherwise. One family of possibilities for the joint density is $f(x,y) = 1 + g(x) h(y)$ for $0 < x < 1$, $0 < y < 1$, 0 otherwise, for functions $g$ and $h$ such that $\int_0^1 g(x)\, dx = \int_0^1 h(y)\, dy = 0$, $-1 \le g(x) \le 1$ and $-1 \le h(y) \le 1$. And there are infinitely many other possibilities.

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There will be many different distributions with the same marginal distributions, so one needs to select a specific way to aggregate the marginal distributions into joint distributions. Assuming they are independent is essentially making one of these possible choices. The most common way to make the choice, is by working with a copula.

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    Michael: Right. In other words, one wants to decompose the joint distribution of (X,Y) into the marginals of$X$and$Y$plus a copula. A move keeping the former fixed while changing the latter is obviously more adapted to this than to the decomposition into the marginal distribution of$X$plus the conditional distribution of$Y$conditionally on X. Thanks for your explanations.2012-01-01