Chapter 7, Problem 7.45P

Let $X_1,X_2,\mathrm{...},X_n$ be independent random variables having an unknown continuous distribution function F. and let $Y_1,Y_2,\mathrm{...},Y_m$ be independent random variables having an unknown continuous distribution function G. Now order those $n+m$ variables, and let $I_i=\{\begin{array}{cc}1& \begin{array}{l}\text{if the }i\text{th smallest of the }n+m\\ \text{variables is from the }X\text{ sample}\end{array}\\ 0& \text{otherwise}\end{array}$

The random variable $R={\displaystyle \underset{i=1}{\overset{n+m}{\sum }}i{I}_i}$ is the sum of the ranks of the X sample and is the basis of a standard statistical procedure (called theWilcoxon 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 3e.