Let $(X,Y)$ be a random vector. Suppose that the marginal distribution functions of $X$ and $Y$ are known (say $F_1$ and $F_2$). Then the joint law of $(X,Y)$ is given by the following formula:

$$F_{X,Y}(x,y)=C(F_1(x),F_2(y)),$$

where C is some copula function. It means, that if the distributions of $X$ and $Y$ are given, we can construct (at least one) joint distribution of $(X,Y)$ conforming with given distributions (e.g. one can set $F_{X,Y}(x,y)=F_1(x)F_2(y)$.

I wonder whether is it possible to extent this property to the case when the distribution of $X+Y$ is known as well (say $F_3$).

My question is: if the distributions of $X$, $Y$ and $X+Y$ are given how can I construct (at least one) joint distribution of $(X,Y)$, conforming with univariate distributions?

Is there any closed-form solution like we have in case when only the marginals are given?

Thank you for the answers!

1more comment