Example 4.18 Simple Investment Portfolio (Means and Variances, Functions of Random Variables) An investor has $1,000 to invest and two investment opportunities, each requiring a minimum of $500. The profit per $100 from the first can be represented by a random variable X, having the following probability distributions: P(X = -5) = 0.4 and P(X = 20 ) = 0.6 The profit per $100 from the second is given by the random variable Y, whose probabil- ity distributions are as follows: P(Y = 0) = 0.6 and P(Y = 25) = 0.4 Random variables X and Y are independent. The investor has the following 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. Solution Random variable X has mean MX = E[X] = ExP(x) = (-5)(0.4) + (20 )0.6) = $10 and variance ox = E[(X - My )?] = >(x - My)2P(x) = (-5-10)2(0.4) + (20 - 10)2(0.6) = 150 Random variable Y has mean My = E[Y] = ZyP(y) = (0)(0.6) + (25 )(0.4) = $10 and variance of = E[(Y - MY)=] = >(y - MY)=P(y) = (0-10)2(0.6) + (25 - 10)2(0.4 ) = 150