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So if I have the random variables X and Y that both assumes the sames values and I know the probability for each value $P_X(x)$ and $P_Y(y)$, how can I build the joint probability distribution for the discrete case?

I`ve tried using $$p_X(x) = \sum_{y} p(x, y) $$ to build a linear system and use the symmetry to solve it, but it did not work.

Can anybody help? Thanks!

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The joint distribution is not determined from the marginals. There's an infinite set of solutions.

Consider the following example of two binary variables $X$ and $Y$ whose joint distribution $p(x,y)$ is as follows: $$ \begin{split} p(0,0) &= 0.2 \\ p(1,0) &= 0.4 \\ p(0,1) &= 0.1 \\ p(1,1) &= 0.3 \end{split} $$ As you can find, the marginal distributions are $p_X(x=1) = 0.7$ and $p_Y(y=1) = 0.4.$

However, those marginals are also true of the joint distribution $q(x,y) = p_X(x) p_Y(y).$ As you can easily see, $q(x,y) \ne p(x,y)$ but they share the same marginal distributions of $X$ and $Y.$

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  • $\begingroup$ Basically, a joint distribution contains information on the marginal distributions and the dependency of the random variables. $~$ So, since they lack that information on dependency, a pair of marginal distributions alone is insufficient to reconstruct their variables' joint distribution. $\endgroup$ – Graham Kemp Jun 15 '17 at 1:46

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