The Wilcoxon rank-sum statistic can be represented as W = R₁ + R₂ + R_m, where R_i is the rank of X_i − Δ₀ among all m + n such differences. When H₀ is true, each R_i is equally likely to be one of the first m + n positive integers; that is, R_i has a discrete uniform distribution on the values 1, 2, 3, …, m + n.

a. Determine the mean value of each R_i when H₀ is true and then show that the mean value of W is m(m + n + 1)/2. The sum of the first k positive integers is k(k + 1)/2.

b. The variance of each R_i is easily determined. However, the R_i’s are not independent random variables because, for example, if m = n = 10 and we are told that R₁ = 5, then R₂ must be one of the other 19 integers between 1 and 20. However, if a and b are any two distinct positive integers between 1 and m + n inclusive, it follows that P(R_i = a and R_j = b) = 1/[(m + n) (m + n – 1)] since two integers are being sampled without replacement from among 1, 2,…, m + n. Use this fact to show that Cov(R_i, R_j) = –(m + n + 1)/12, and then show that the variance of W is mn(m + n + 1)/12.