# Joint Probability From Correlation Function

Dear all

I have the following problem. Given a set of correlated binary variables, can I determine the joint probability from the correlation function?

{Xi} is a set of binary variables
Pr(Xi=1) = p and Pr(Xi=0) = q for all i
Corr(Xi Xj) = cij

cij is symmetric

Now how can I determine the joint probability Pr({Xi, Xj, Xk ....})
For the joint probability of two variables I think I have the answer.
Noting that cij= (E(Xi Xj) - p2) / pq, and using the notation {Xi=xi,Xj=xi} -> {xi,xj}

I have
Pr( {1,1} )= E(Xi Xj) = p*q*cij + p2

and by symmetry
Pr( {0,0} ) = p*q*cij + q2

and Pr( {0,1} ) = Pr( {1,0} ) = ( 1-Pr({1,1})-Pr({0,0}) ) / 2 = p*q*(1- cij )
using that p+q = 1

How can I proceed to get Pr( {Xi,Xj, Xk} ), and generally Pr( {Xi,Xj, Xk, ….} )? I'm I missing something obvious? Any help or input is highly appreciated.

best
t

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## Answers and Replies

You will need higher moments - the pairwise correlations are not enough for n>2. Look into Teugels (1990) ‘Some Representations of the Multivariate Bernoulli and Binomial Distributions’, where he provides a formulation for a general multivariate Bernoulli with dependencies, for n dimensions.

Omri.