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 2.5, then the ranks of positive observations are 2 and 3, so W
5. In Chapter 15, W will be called Wilcoxon’s signed-rank statistic. W can be represented as follows:

W
1
Y1 2
Y2 3
Y3 . . . n
Yn

n i
1 i
Yi where the Yis 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.

[Hint: The first n positive integers sum to n(n 1)/2.]

b. Determine V(Yi

) and then V(W) [Hint: The sum of the squares of the first n positive integers can be expressed as n(n 1)(2n 1)/6.]