76.

a. Use the general formula for the variance of a linear combination to write an expression for V(aX  Y). Then let a  sY /sX, and show that r  1. [Hint: Variance is always  0, and Cov(X, Y)  sX # sY # r.]

b. By considering V(aX  Y), conclude that r 1.

c. Use the fact that V(W) 0 only if Wis a constant to show that r  1 only if Y  aX  b.

v77. A rock specimen from a particular area is randomly selected and weighed two different times. Let W denote the actual weight and X1 and X2 the two measured weights. Then X1WE1 and X2WE2, where E1 and E2 are the two measurement errors.

Suppose that the Ei s are independent of one another and of W and that V(E1)  V(E2)  .

a. Express r, the correlation coef cient between the two measured weights X1 and X2, in terms of

, the variance of actual weight, and , the variance of measured weight.

b. Compute r when sW  1 kg and sE  .01 kg.