Let X1,X2,...,Xn be independent random variables having an unknown continuous distribution function E, 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.
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