Consider a random sample of size n from a continuous dis- tribution 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: = W1Y+2 Y+3 Y++n. Y 11 - i-1 3 where the Y's are independent Bernoulli rv's, each with p 5 (Y 1 corresponds to the observation with rank i being positive).

a. Determine E(Y) 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) 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.]