Determining discrete probabilities for X and Y given their correlation coefficient.

Let XXX and YYY be two stochastic variables which can take the values 00, 11 and 22. Suppose I know the value of their correlation coefficient, what are some good methods to approximate a set of probabilities for XX and YY?

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 XX and YY have, so I don’t see an obvious way to apply the statistical methods I’ve been taught.

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The correlation between XX and YY is not enough to determine a joint probability distribution. More information is needed.
– Math1000
2 days ago

@Math1000 I suspected as much. However, that’s all the information I have. I’m not really looking for a distribution, but rather just the joint probabilities which gives the known correlation. Since there is a lack of information here, I guess the best way to “choose” the joint probabilities would be a numerical approximation?
– Auclair
2 days ago

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