Let X1, X2, . . . ,Xn be independent random variables having an unknown continuous distribution function F, and let Y1,Y2, . . . ,Ym be independent random variables having an unknown continuous distribution function G. Now order those n + m variables, and let

The random variable

is the sum of the ranks of the X sample and is the basis of a standard statistical procedure (called the Wilcoxon sum-of-ranks test) for testing whether F and G are identical distributions. This test accepts the hypothesis that F = G when R is neither too large nor too small. Assuming that the hypothesis of equality is in fact correct, compute the mean and variance of R.
Use the results of Example 3e.

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1 if the ith smallest of the n m Iivariables is from the X sample 0 otherwise