Correlation of two weighted dice combinations

Question

Roll two fair dice; let X and Y be the results. Define A = 2X + Y and B = X - Y. What is the covariance of A and B, and their correlation?

Accepted Answer

Write \sigma^2 = \mathrm{Var}(X) = \mathrm{Var}(Y) = 	frac{35}{12} (the two dice are identically distributed), and \mathrm{Cov}(X, Y) = 0 by independence. Covariance. Expand every cross term: \mathrm{Cov}(A, B) = \mathrm{Cov}(2X + Y,\, X - Y) = 2\,\mathrm{Var}(X) - 2\,\mathrm{Cov}(X,Y) + \mathrm{Cov}(Y,X) - \mathrm{Var}(Y). The two covariance terms vanish, leaving 2\sigma^2 - \sigma^2 = \sigma^2 = 	frac{35}{12}. Variances. \mathrm{Var}(A) = \mathrm{Var}(2X + Y) = 4\sigma^2 + \sigma^2 = 5\sigma^2, \qquad \mathrm{Var}(B) = \mathrm{Var}(X - Y) = \sigma^2 + \sigma^2 = 2\sigma^2. Correlation. …

Correlation of two weighted dice combinations

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