Given two marginal distributions are they consistent with an underlying joint distribution?

Question

Given two random probability distributions how can we check if they are consistent with an underlying joint probability distribution?

Specifically, take two random variables $X$ and $Y$. If they are the marginals of some underlying distribution then they should satisfy the following equations, $$ Pr(X = x_i) = \sum_j Pr(X= x_i, Y = y_j) = \sum_j Pr(X = x_i | Y = y_j)~ Pr(Y = y_j)$$

$$ Pr(Y = y_j) = \sum_i Pr(X= x_i, Y = y_j) = \sum_i Pr(Y = y_j | X = x_i) ~Pr(X = x_i)$$

Under what conditions will the above equations be satisfied?

Answer 1

Any two distributions can be the marginals of a joint distribution. Simply choose the joint distribution such that the marginals are independent.