A couple of properties of correlation

Spotted these in Langford, E.; Schwertman, N. & Owens, M. (2001) [Is the Property of Being Positively Correlated Transitive? The American Statistician, 55, 322-325.]

1. Let U, V, and W be independent random variables. Define X = U+V, Y = V+W, and Z = WU. Then the correlation between X and Y is positive, Y and Z is positive, but the correlation between X and Z is negative.

It’s easy to see why.  X and Y are both V but with different uncorrelated noise terms. Y and Z have W in common, again with different noise terms. Now X and Z have U in common: for this pair, X is U plus some noise and Z is –U plus some noise which is uncorrelated with the noise in X.

2. If X, Y, and Z are random variables, and X and Y are correlated (call the coefficient \(r_1\)), Y and Z are correlated (\(r_2\)), and \(r_1^2 + r_2^2 > 1\), then X and Z are positively correlated.