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The random variables X and Y are independent.

The investor has the following three possible strategies:

(a) $ 1,000 in the first investment

(b) $1,000 in the second investment

(c) $500 in each investment

Find the mean and variance of the profit from each strategy.

The random variable X has mean fix = E(X) = Ejt/Mjc) = (—5)(.4) + (20) (.6) = $10

= (-5 - 10)2(.4) 4- (20 - 10)2(.6) = 150 Hence, the mean profit from strategy (a) is

and the variance is

Var(lOX) = 100 Var(X) = 15,000 The random variable Y has mean

= (0 - 10)2(.6) + (25 - 10)2(.4) = 150 Therefore, strategy (b) has mean profit

and variance

Var(lOy) = 100 Var(T) - 15,000 Now, the return from strategy (c) is 5X + 5y, which has mean

E(5X + 5Y) = E(5X) + E(5Y) = 5 E(X) + 5 E(Y) = $100

Thus, all three strategies have the same expected profit. However, since X and Y are independent, and hence have covariance 0, the variance of the return from strategy (c) is

Var(5X + 5Y) = Var(5X) + Var(5 Y) = 25ax2 + 25a-,2 = 7,500

This is smaller than the variances of the other strategies, reflecting the decrease in risk that follows from diversification in an investment portfolio. This investor should certainly prefer strategy (c), since it yields the same expected return as the other two, but with lower risk.

In the remaining sections in this chapter, we will discuss some specific discrete probability distributions that have important applications in the business area.

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