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I'm looking for a "nice" way to parametrize the joint distribution of multiple, possibly correlated discrete random variables on {0,1}. Even for N=2, there doesn't seem to be an obvious way to do it.

The kind of thing I'm looking for, is how bi-normal distributions can be simply parametrized by the means and variances of the individual distributions and by a single "correlation" parameter. Things get more complicated in higher dimensions of course, but is there such a nice analog in my case?

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The probabilities on $\{0,1\}^N$ have to add up to $1$, so they give the barycentric coordinates of a point in a simplex of dimension $2^N-1$. For $N=2$, you can use other coordinates, including triples

$(E(X),E(Y), E(XY))$

or

$(E(X),E(Y), \text{cov}(X,Y))$

where $\text{cov}(X,Y)=E(XY)-E(X)E(Y)) = \rho(X,Y)\sigma(X)\sigma(Y)$.

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