39. Let X1, X2, X be independent random variables having an unknown continuous distribution function F, and let Y1, Y2, Y be independent random variables having an unknown continuous distribution function G.

Now order those + variables and let

$$I_i= \begin{cases}

1 & \text{if the ith smallest of the } n+m \text{ variables is from the } X \text{ sample}\\

0 & \text{otherwise}

\end{cases}$$

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The random variable R = +
Σ
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.
HINT: Use the results of Example 3d.