SUM across to get the values, that is where the name marginal comes from.
Do that across and down (for x's distribution).
IN order for these two rvs to be independent... for EVERY position the sum across and the sum down's product must equal the probability of that position.
The shortcut formula for covariance is and if the rvs are indep, this is zero.
The definition of correlation is and if the rvs are indep, this is zero.