Chapter 5.5, Problem 70E

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, +12.1, 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 + Y₂ + 3. Y3 +….+ n Y_n = ${\displaystyle \underset{i=1}{\overset{n}{\sum }}i\cdot Y_i}$

where the Y_i’s are independent Bernoulli rv’s, each with p = .5 (Y_i = 1 corresponds to the observation with rank i being positive).

a. Determine E(Y_i) and then E(W) using the equation for W. [Hint: The first n positive integers sum to n(n + 1)/2.]

b. Determine V(Y_i) 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.]