Problem 2 (28 points) Suppose that 21, 22, 23 are random variables with IE(zi) = 0, var(zi) = 1; and, cov(zi,zj) = p, for 1' 7E j. Letyi =Z1 Zg+1

Problem 2 (28 points)

Suppose that z1, z2, z3 are random variables with

E(zi) = 0, var(zi) = 1; and, cov(zi,zj) = ρ, for i ̸= j. Lety1 =z1−z2+1andy2 =z1+z2−2z3+2.

1. (a) Show that y1 and y2 are uncorrelated. [Hint: use matrix techniques.]
2. (b) Are y1 and y2 independent? Briefly explain why.
3. (c) Find the variance of (y1 + y2)/2.
4. (d) Someone claims that (y1 + y2)/2 is the projection of [y1, y2]T onto the subspace spanned by [1, 1]T . Is that true? Briefly explain.