Page 108 -

Exercises 107

a. Explain how the marginal probability that Y = y can be calculated
from the joint probability distribution. [Hint: This is a generalization
of Equation (2.16).]

b. Show that E(Y) = E[E(Y 0 X, Z)]. [Hint: This is a generalization of

Equations (2.19) and (2.20).]

2.21 X is a random variable with moments E(X), E(X2), E(X3), and so forth.

a. Show E(X - m)3 = E(X3) - 3[E(X2)][E(X)] + 2[E(X)]3.

b. Show E(X - m)4 = E(X4) - 4[E(X)][E(X3)] + 6[E(X)]2[E(X2)] -
3[E(X)]4.

2.22 Suppose you have some money to invest—for simplicity, $1—and you are
planning to put a fraction w into a stock market mutual fund and the rest,
1 - w, into a bond mutual fund. Suppose that $1 invested in a stock fund
yields Rs after 1 year and that $1 invested in a bond fund yields Rb, suppose
that Rs is random with mean 0.08 (8%) and standard deviation 0.07, and
suppose that Rb is random with mean 0.05 (5%) and standard deviation
0.04. The correlation between Rs and Rb is 0.25. If you place a fraction w
of your money in the stock fund and the rest, 1 - w, in the bond fund, then
the return on your investment is R = wRs + (1 - w)Rb.

a. Suppose that w = 0.5. Compute the mean and standard deviation of R.
b. Suppose that w = 0.75. Compute the mean and standard deviation of R.
c. What value of w makes the mean of R as large as possible? What is

the standard deviation of R for this value of w?

d. (Harder) What is the value of w that minimizes the standard deviation
of R? (Show using a graph, algebra, or calculus.)

2.23 This exercise provides an example of a pair of random variables X
and Y for which the conditional mean of Y given X depends on X but
corr(X, Y) = 0. Let X and Z be two independently distributed standard
normal random variables, and let Y = X2 + Z.

a. Show that E(Y 0 X ) = X2.

b. Show that mY = 1.

c. Show that E(XY ) = 0. (Hint: Use the fact that the odd moments of a
standard normal random variable are all zero.)

d. Show that cov(X, Y ) = 0 and thus corr(X, Y ) = 0.