Let $X$ and $Y$ be two stochastic variables which can take the values $0$, $1$ and $2$. Suppose I know the value of their correlation coefficient, what are some good methods to approximate a set of probabilities for $X$ and $Y$?
I tried to impose certain restrictions on the probabilities, such as certain of them being equal (I even went so far once as to assign values to the marginal probabilities) but I'm not really getting anywhere.
Would this be a good time to write a program which approximates them? There are probably much better methods to do this, but the problem is I don't know which distribution $X$ and $Y$ have, so I don't see an obvious way to apply the statistical methods I've been taught.