# Question

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.

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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