# Building joint probability distribution [closed]

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!

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 $Y.$
• 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. – Graham Kemp Jun 15 '17 at 1:46