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Let (S₁, ... , S_n ) be a vector of ranks that is uniformly distributed over the set of all n! permutations of (1, 2, ... , n). Show that the joint marginal probability distribution of (S_i , S_j), for i ≠ j = 1, ... , n, is given by

Use this fact to show that cov(S_i , S_j) = -(n + 1)/12, for i ≠ j = 1, ... , n.

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