Let X1,X2, . . . ,Xn be independent random variableshaving

Chapter 7, Problem 7.43

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Let X1,X2, . . . ,Xn be independent random variableshaving an unknown continuous distributionfunction F, and let Y1,Y2, . . . ,Ym be independentrandom variables having an unknown continuousdistribution function G. Now order those n + mvariables, and letIi =1 iftheith smallest of the n + mvariables is from the X sample0 otherwiseThe random variable R =n+mi=1iIi is the sum of theranks of the X sample and is the basis of a standardstatistical procedure (called the Wilcoxon sum-ofrankstest) for testing whether F and G are identicaldistributions. This test accepts the hypothesisthat F = G when R is neither too large nor toosmall. Assuming that the hypothesis of equality isin fact correct, compute the mean and varianceof R.Hint: Use the results of Example 3e.