Dear all(adsbygoogle = window.adsbygoogle || []).push({});

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

{X_{i}} is a set of binary variables

Pr(X_{i}=1) = p and Pr(X_{i}=0) = q for all i

Corr(X_{i}X_{j}) = c_{ij}

c_{ij}is symmetric

Now how can I determine the joint probability Pr({X_{i}, X_{j}, X_{k}....})

For the joint probability of two variables I think I have the answer.

Noting that c_{ij}= (E(X_{i}X_{j}) - p^{2}) / pq, and using the notation {X_{i}=x_{i},X_{j}=x_{i}} -> {x_{i},x_{j}}

I have

Pr( {1,1} )= E(X_{i}X_{j}) = p*q*c_{ij}+ p^{2}

and by symmetry

Pr( {0,0} ) = p*q*c_{ij}+ q^{2}

and Pr( {0,1} ) = Pr( {1,0} ) = ( 1-Pr({1,1})-Pr({0,0}) ) / 2 = p*q*(1- c_{ij})

using that p+q = 1

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

best

t

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Joint Probability From Correlation Function

**Physics Forums | Science Articles, Homework Help, Discussion**