Mathematics · Nonparametric Statistics

Let X 1 , X 2 , ..., X n and Y 1 , Y 2 , ..., Y m be two independent random samples from a common continuous distribution F. Let W be the sum of ranks of X observations in the combined ranking of all the N = n + m observations. Which of the following statements are true? I. E(W) = n(N + 1) 2 II. E(W) = N(N + 1) 4 III. The distribution of W is symmetric about n(N + 1) 2 IV. The distribution of W does not depend on F