Chapter 7, Problem 7.44P

Between two distinct methods for manufacturing certain goods, the quality of goods produced by method i is a continuous random variable having distribution $F_i,i=1,2$. Suppose that n goods are produced by method 2 and m by method 2. Rank the $n+m$ goods according to quality, and let $X_j=\{\begin{array}{c}1\text{if the i th best was produced from method 1}\\ \text{2 }\text{otherwise}\end{array}$ $X_1,X_2,\mathrm{...}{X}_{n+m}$, which consists of n, l’s and m 2’s, let R denote the number of runs of 1. For instance, if $n=5,m=2$, and $X=1,2,1,1,1,1,2$ ,then $R=2$. lf $F_1=F_2$ (that is. if the two methods produce identically distributed goods), what are the mean and variance of R?