Are the sum and difference of two dice independent?

Question

Roll two fair dice; let X and Y be the results. Define S = X + Y and D = X - Y. What is the covariance of S and D? Are they independent?

Accepted Answer

Covariance. By bilinearity, with X, Y independent (so \mathrm{Cov}(X,Y) = 0): \mathrm{Cov}(S, D) = \mathrm{Cov}(X+Y,\, X-Y) = \mathrm{Var}(X) - \mathrm{Cov}(X,Y) + \mathrm{Cov}(Y,X) - \mathrm{Var}(Y) = \mathrm{Var}(X) - \mathrm{Var}(Y). Since X and Y are identically distributed, \mathrm{Var}(X) = \mathrm{Var}(Y) and \mathrm{Cov}(S, D) = 0. S and D are uncorrelated. Independence? No. One counterexample suffices: if S = 12 then both dice are sixes, forcing D = 0: P(D = 0 \mid S = 12) = 1 
e P(D = 0) = \frac{6}{36} = \frac16. Knowing the sum changes the distribution of the difference (e.g. extr…

Are the sum and difference of two dice independent?

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