2.13 More on variances and covariances This problem presents a few useful mathematical facts about variances and covariances.

a. Show that Var(x) =E(x)-[E(x). ===

b. Show that the result in part

(a) can be generalized as Cov(x,y) = E(xy) - E(x)E(y). Note: If Cov(x, y) = 0, then E(xy) = E(x)E(y).

c. Show that Var(ax + by) a Var(x) + b Var(y) 2ab Cov(x,y). =

d. Assume that two independent random variables, x and y, are characterized by E(x) = E(y) and Var(x) Var(y). Show that E(0.5x+0.5y) E(x). Then use part

(c) to show that Var(0.5x +0.5y) 0.5 Var(x). Describe why this fact provides the rationale for diversification of assets.