1. In a lot of n components, 30% are defective. Two components are drawn at random and tested. Let A be the event that the first component drawn is defective,….

## Find the value of K that minimizes the risk in part (c).

1. Refer to Example 2.54.

a. Find Cov(X,Y).

b. Find ρX,Y .

2. Refer to Exercise 18.

a. Find Cov(X, Y).

b. Find ρX,Y .

c. Find the conditional probability density function fY|X (y | 0.5).

d. Find the conditional expectation E(Y | X = 0.5).

3. An investor has $100 to invest, and two investments between which to divide it. If she invests the entire amount in the first investment, her return will be X, while if she invests the entire amount in the second investment, her return will be Y. Both X and Y have mean $6 and standard deviation (risk) $3. The correlation between X and Y is 0.3.

a. Express the return in terms of X and Y if she invests $30 in the first investment and $70 in the second.

b. Find the mean return and the risk if she invests $30 in the first investment and $70 in the second.

c. Find the mean return and the risk, in terms of K, if she invests $K in the first investment and $(100 − K) in the second.

d. Find the value of K that minimizes the risk in part (c).

e. Prove that the value of K that minimizes the risk in part (c) is the same for any correlation ρX,Y with ρX,Y ≠ 1.