How correlated is one die with the total?

Question

Roll two fair dice; let X and Y be the two results and S = X + Y the sum. What is the correlation between a single die X and the sum S?

Accepted Answer

Covariance. By bilinearity, with X and Y independent so \mathrm{Cov}(X, Y) = 0: \mathrm{Cov}(X, S) = \mathrm{Cov}(X, X + Y) = \mathrm{Var}(X) + \mathrm{Cov}(X, Y) = \mathrm{Var}(X). One die has \mathrm{Var}(X) = 	frac{35}{12}. Variances. \mathrm{Var}(S) = \mathrm{Var}(X) + \mathrm{Var}(Y) = 2 \cdot 	frac{35}{12} = 	frac{35}{6}. Correlation. ho = \frac{\mathrm{Cov}(X, S)}{\sqrt{\mathrm{Var}(X)\,\mathrm{Var}(S)}} = \frac{35/12}{\sqrt{	frac{35}{12} \cdot 	frac{35}{6}}} = \frac{35/12}{\,35 / (6\sqrt{2})\,} = \frac{1}{\sqrt{2}} \approx 0.707. The clean answer 	frac{1}{\sqrt2} has a tidy in…

How correlated is one die with the total?

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