Given that random variables $X$ and $Y$ are positively correlated, and random variables $Z$ and $W$ are also positively correlated, can it be concluded that $X+Z$ is positively correlated with $Y+W$? Is there a theorem or proposition that supports this conclusion?
2026-03-29 12:03:51.1774785831
Additivity of Correlation
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Hi: Following what Kurt G started: Cov(X + Z, Y + W) = Cov(X,Y) + Cov(X,W) + Cov(Z,Y) + Cov(Z,W). So, the first and the last term are positive but you don't know about the second and third terms. It's possible that they could be negative and cancel out the positive terms to make the overall covariance zero.