[math-fun] Random variable puzzle

Suppose X, Y, Z, W are real random variables with a joint distribution such that each one has a finite mean and standard deviation. Suppose that all pairs of these random variables have the same correlation coefficient:* R = rho(X,Y) = rho(X,Z) = rho(X,W) = rho(Y,Z) = rho(Y,W) = rho(Z,W) . Find the minimum possible value of R. —Dan ____________________________________ * The definition of the correlation coefficient rho(U,V) of two random variables U and V is the expected product of their standardizations: rho(U,V) = E( ((U-mu_U)/sigma_U) * ((V-mu_V)/sigma_V)) ) where E is expectation, mu denotes mean and sigma denotes standard deviation.