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Having the marginal distributions, say $f(x)$ and $f(y)$, how would we get the conditional distribution $f(x|y)$? The relation is given by: $$f(x)=\int f(x|y)f(y)dy$$ Do we need to find the derivatives of both sides and then solve a differential equation?!

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The marginals don't determine the conditional.

Indeed copulas are a way to have all manner of different joint distributions with given marginals

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