pr.probability - Nice Way to Parametrize a bunch of non-independent discrete random variables

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