39. Suppose we take 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. We now disregard the signs of the observations, rank them from smallest to largest in absolute value, and then 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, soW5. In Chapter 14,Wwill be calledWilcoxon’s signed-rank statistic. W can be represented as follows:

 a n

i1 i # Yi W  1 # Y1  2 # Y2  3 # Y3  . . .  n # Yn where the Yi s are independent Bernoulli rv s, each with p  .5 (Yi  1 corresponds to the observation with rank i being positive). Compute the following:

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

b. V(Yi) and thenV(W) [Hint: The sum of the squares of the rst n positive integers is

(2n  1)/6.]