Covariance of two blends when the base pair is negatively linked

Question

Two ingredients have variances \mathrm{Var}(X) = 2 and \mathrm{Var}(Y) = 6, with covariance \mathrm{Cov}(X, Y) = -1. Two blends are U = 3X - Y, \qquad V = X + Y. What is the covariance of U and V?

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Expand the covariance of the two blends into cross-pieces, pull out constants, and convert same-variable pieces into variances. The extra care here is the negative covariance of the base pair. \mathrm{Cov}(3X - Y,\; X + Y) = \mathrm{Cov}(3X, X) + \mathrm{Cov}(3X, Y) + \mathrm{Cov}(-Y, X) + \mathrm{Cov}(-Y, Y). Rewrite each piece: = 3\,\mathrm{Var}(X) + 3\,\mathrm{Cov}(X, Y) - \mathrm{Cov}(X, Y) - \mathrm{Var}(Y). Plug in, remembering \mathrm{Cov}(X, Y) = -1: = 3(2) + 3(-1) - (-1) - 6 = 6 - 3 + 1 - 6 = -2. Takeaway. The expansion handles negative covariances automatically as long as you carry …

Covariance of two blends when the base pair is negatively linked

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