
Joint Probability
I have a question for you and thank you in advance for your answers and ideas.
Let us suppose that we have the marginal distributions of two r.v X and Y, and also the law of XY (or any measurable/linear function of X and Y).
How could we find the joint distribution f(x,y) (not always unique) of both my r.v?

I'd advise you to look up copulas.
A bunch oof my colleagues worked on that topic for 30 plus years now, including Abe Sklar.
http://en.wikipedia.org/wiki/Copula_(statistics)