1. Expected value

2. Question 1 You pay your sales personnel a commission of 75% of the amount they sell over $2000. X = Sales has mean $5000 and standard deviation $1000. What are mean and standard deviation of pay?

3. Question 1 ( X - 2000) represents the basis for the commission and "Pay" is 75% of that Pay = (0.75)( X - 2000) = 0.75 X - 1500 E [Pay] = E[ 0.75 X - 1500 ] = 0.75 E [ X ] - 1500 = 0.75(5000) - 1500 = $2250

4. Question 1 ( X - 2000) represents the basis for the commission and "Pay" is 75% of that Pay = (0.75)( X - 2000) = 0.75 X - 1500  [Pay] =  [ 0.75 X - 1500 ] = 0.75  [ X ] = 0.75(1000) = $750 Note: We are not combining so we don ’ t need to use variance.

5. Question 2 - The Portfolio Affect You are considering purchase of stock in two different companies, X and Y. Return after one year for stock X is a random variable with  X = $112,  X = 10. Return for stock Y (a different company) has the same  and . Assuming that X and Y are independent, which portfolio has less variability, 2 shares of X or one each of X and Y ?

6. Question 2 - The Portfolio Affect The returns from 2 shares of X will be exactly twice the returns from one share, or 2X. The returns from one each of X and Y is the sum of the two returns, X+Y.

7. Question 2 - The Portfolio Affect E [ aX + b ] = a E [ X ] + b E [ aX + bY ] = aE [ X ] + bE [ Y ] E [2 X + 0] = 2(112) = 224 E [ X + Y ] = 112 + 112 = 224

8. Question 2 - The Portfolio Affect = 2(10) = 20 =14.14

9. Conclusion : X + Y has smaller standard deviation than 2 X.

10. Insight : Why does X + Y have a narrower probability distribution than 2 X ? Since X and Y vary independently, losses in one are sometimes offset by gains in the other. With 2 shares of stock of the same company, losses and gains are just doubled. This is one version of the old saying, "Don ’ t put all of your eggs into one basket!"

11. Question 3 In what interval will the return of a portfolio consisting of 2 units of stock X and 3 units of stock Y occur 2/3 of the time, according to the empirical rule? (Use X and Y from question 2 i.e.  X = $112,  X = 10..)

12. Question 3 First, use (3a) to get the mean:

13. Next, use "special case" formula (3b) to get the standard deviation:

14. The empirical rule states that "approximately 2/3 of the time, a random variable will be within 1  of its mean". Here this interval is $560  36.06.

15. Question 4 The selling price of a product is $30, but it costs the seller $20. The forecast of the number of units that will be sold in the upcoming month is 5000, with standard deviation 100. The seller has a fixed cost of $8,000 per month. In what interval will net profits lie for the upcoming month, with 95% probability, according to the empirical rule? The empirical rule states that "approximately 95% of the time, a random variable will be within 2  of its mean".

16. Question 4 Let X = number of units sold next month. Profits = (30 - 20) X - 8000 = 10 X - 8000. Expected profits = E [10 X - 8000] = 10(5000) - 8000 = $42,000. Standard deviation of profits =  [10X - 8000] = 10  X = 10(100) = $1000. The empirical rule states that "approximately 95% of the time, a random variable will be within 2  of its mean" so the 95% range for returns is $42,000  2(1,000).