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Expected value

Expected value. 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?. Question 1. ( X - 2000) represents the basis for the commission

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Expected value

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  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

  8. Question 2 - The Portfolio Affect

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

  10. Insight: • Why does X+Y have a narrower probability distribution than 2X? • 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 = 10X - 8000. • Expected profits = E[10X - 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).

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