Consider a random sample of size n from a continuous distribution having median 0 so tha...

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:

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. [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.]