Covariance of two blends of the same two signals

Question

Two base signals have variances \mathrm{Var}(X) = 4 and \mathrm{Var}(Y) = 9, with covariance \mathrm{Cov}(X, Y) = 2. You build two new quantities from them: U = 2X + Y, \qquad V = X - Y. What is the covariance of U and V?

Accepted Answer

Covariance spreads across sums the way multiplying-out a product does: to find the covariance of two blends, expand term by term, and each piece becomes a covariance of the individual parts. The rules you need are that constants pull out front, and the covariance of a variable with itself is its variance. Expand \mathrm{Cov}(2X + Y,\; X - Y) into the four cross-pieces: \mathrm{Cov}(2X, X) + \mathrm{Cov}(2X, -Y) + \mathrm{Cov}(Y, X) + \mathrm{Cov}(Y, -Y). Pull the constants out and rewrite each piece using variances and covariances: = 2\,\mathrm{Var}(X) - 2\,\mathrm{Cov}(X, Y) + \mathrm{Cov}(X…

Covariance of two blends of the same two signals

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