Consider a random sample of size n from a continuous distribution having median 0 so that the probability of any one observation being positive is .5. Disregarding the signs of the observations, rank them from smallest to largest in absolute value, and let W = the sum of the ranks of the observations having positive signs. For example, if the observations are -.3, + .7, + 2.1, and - 2.=, then the ranks of positive observations are 2 and 3, so W = 5. In Chapter 1=, W will be called Wilcoxon's signed-rank statistic. W can be represented as follows:

where the Yi's are independent Bernoulli rv's, each with p = .5 (Yi = 1 corresponds to the observation with rank i being positive).
a. Determine E(Yi) and then E(W) using the equation for W.
b. Determine V(Yi) and then V(W).

w = 1. Y, + 2 Y, + 3 Y, + . - + n -i.. Y, i-1