# 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* = *W*–*U*. 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 ), *Y* and *Z* are correlated (), and , then *X* and *Z* are positively correlated.

And… hmm… I’m not sure why this holds.