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?