I need a formula for the probability of an event in a n-variate Bernoulli distribution X∈{0,1}n with given P(Xi=1)=pi probabilities for a single element and for pairs of elements P(Xi=1∧Xj=1)=pij. Equivalently I could give mean and covariance of X.

I already learned that there exist many {0,1}n distributions having the properties just as there are many distributions having a given mean and covariance. I am looking for a canonical one on {0,1}n, just as the Gaussian is a canonical distribution for Rn and a given mean and covariance.

**Answer**

See the following paper:

J. L. Teugels, Some representations of the multivariate Bernoulli and binomial

distributions,Journal of Multivariate Analysis, vol. 32, no. 2, Feb. 1990, 256–268.

Here is the abstract:

Multivariate but vectorized versions for Bernoulli and binomial distributions are established using the concept of Kronecker product from matrix calculus. The multivariate Bernoulli distribution entails a parameterized model, that provides an alternative to the traditional log-linear model for binary variables.

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