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!


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 $Y.$

  • $\begingroup$ 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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