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Can one determine the joint distribution of $(X,Y)$ from the probability densities of $X$, $Y$, and $X+Y$?

Here, $X$ and $Y$ are random variables from a sample space $(\Omega, \mathbb{P}) \to \mathbb{R}$.

This is NOT a homework question.

Did
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user062295
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    If you have the joint distribution of $X$ and $Y$, you simply need to integrate out $Y$. The joint distribution specifies the marginals... – Batman Apr 09 '17 at 23:22
  • Can you determine the joint distribution of $X$ and $Y$ from the density of $X+Y$ , the probability density of $X$ and the probability density of $Y$? I think this might have been the question. – user062295 Apr 09 '17 at 23:25

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Let $X$ and $Y$ be discrete random variables on $\{1,2,\ldots,n\}.$ Then we have that $X+Y \in \{2,3,\ldots,2n\}.$ Now, all the information from the distributions we know, can be written in terms of linear equations. We know that we have $n^2$ unknowns, so the question is do we have at least $n^2$ equations to work with. The answer is, for large enough $n$, we do not. We can think about each mass of $X+Y$ as giving us an equation, which yields $2n-1$ linear equations, then each mass of $X$ and $Y$ gives us an equation, for $2n$ more equations. Additionally, because these are random variables we get an equation from normalization yielding a total of $$2n-1 + 2n + 1 = 4n$$ linear equations. Then, for $n>4$ we certainly cannot uniquely determine the joint distribution from the information of $X,Y,X+Y$.

David
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