Of course variances are in squared units The company would prefer to know stan

Of course variances are in squared units the company

This preview shows page 8 - 9 out of 23 pages.

Of course, variances are in squared units. The company would prefer to know stan- dard deviations, which are in dollars. The standard deviation of the payout for two inde- pendent policies is 1 299,200 = + 546.99. But the standard deviation of the payout for a single policy of twice the size is 1 598,400 = + 773.56, or about 40% more. If the company has two customers, then, it will have an expected annual total payout of $40 with a standard deviation of about $547. For Example ADDING THE DISCOUNTS RECAP: The Valentine’s Day Lucky Lovers discount for couples averages $5.83 with a standard deviation of $8.62. We’ve seen that if the restaurant doubles the dis- count offer for two couples dining together on a single check, they can expect to save $11.66 with a standard deviation of $17.24. Some couples decide instead to get separate checks and pool their two discounts. QUESTION: You and your amour go to this restaurant with another couple and agree to share any benefit from this promotion. Does it matter whether you pay separately or together? ANSWER: Let X 1 and X 2 represent the two separate discounts, and T the total; then T = X 1 + X 2 . E 1 T 2 = E 1 X 1 + X 2 2 = E 1 X 1 2 + E 1 X 2 2 = 5.83 + 5.83 = $11.66, so the expected saving is the same either way. The cards are reshuffled for each couple’s turn, so the discounts couples receive are independent. It’s okay to add the variances: Var 1 T 2 = Var 1 X 1 + X 2 2 = Var 1 X 1 2 + Var 1 X 2 2 = 8.62 2 + 8.62 2 = 148.6088 SD 1 T 2 = 1 148.6088 = $12.19 When two couples get separate checks, there’s less variation in their total discount. The standard deviation is $12.19, compared to $17.24 for couples who play for the double discount on a single check. In general, The mean of the sum of two random variables is the sum of the means. The mean of the difference of two random variables is the difference of the means. If the random variables are independent, the variance of their sum or difference is always the sum of the variances. E 1 X { Y 2 = E 1 X 2 { E 1 Y 2 Var 1 X { Y 2 = Var 1 X 2 + Var 1 Y 2 Wait a minute! Is that third part correct? Do we always add variances? Yes. Think about the two insurance policies. Suppose we want to know the mean and stan- dard deviation of the difference in payouts to the two clients. Since each policy has an expected payout of $20, the expected difference is 20 - 20 = + 0. If we also sub- tract variances, we get $0, too, and that surely doesn’t make sense. Note that if the out- comes for the two clients are independent, the difference in payouts could range from + 10,000 - + 0 = + 10,000 to + 0 - + 10,000 = - + 10,000, a spread of $20,000. The variability in differences increases as much as the variability in sums. If the company has two customers, the difference in payouts has a mean of $0 and a standard deviation of about $547 (again). PYTHAGOREAN THEOREM OF STATISTICS We often use the stan- dard deviation to measure variability, but when we add independent random variables, we use their variances. Think of the Pythagorean Theorem. In a right triangle (only), the square of the length of the hypotenuse is the sum of the squares of the lengths of the other two sides: c 2 = a 2 + b 2 .
Image of page 8
Image of page 9

You've reached the end of your free preview.

Want to read all 23 pages?

  • Winter '15

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture