1. The Variance of a Random Variable Lecture 35 Section 7.5.1 Fri, Mar 26, 2004

2. The Variance of a Discrete Random Variable Variance of a Discrete Random Variable – The square of the standard deviation of that random variable. Variance of a Discrete Random Variable – The square of the standard deviation of that random variable. The variance of X is denoted by The variance of X is denoted by  2 or Var(X) The standard deviation of X is denoted by . The standard deviation of X is denoted by .

3. The Variance and Expected Values The variance is the expected value of the squared deviations. The variance is the expected value of the squared deviations. That agrees with the earlier notion of the average squared deviation. That agrees with the earlier notion of the average squared deviation. Therefore, Therefore, Var(X) = E((X – µ) 2 ).

4. Example of the Variance Again, let X be the number of children in a household. Again, let X be the number of children in a household. x P(X = x) 00.10 10.30 20.40 30.20

5. Example of the Variance Subtract the mean (1.70) from each value of X to get the deviations. Subtract the mean (1.70) from each value of X to get the deviations. x P(X = x) x – µ 00.10-1.7 10.30-0.7 20.40+0.3 30.20+1.3

6. Example of the Variance Square the deviations. Square the deviations. x P(X = x) x – µ (x – µ) 2 00.10-1.72.89 10.30-0.70.49 20.40+0.30.09 30.20+1.31.69

7. Example of the Variance Multiply each squared deviation by its probability. Multiply each squared deviation by its probability. x P(X = x) x – µ (x – µ) 2 (x – µ) 2  P(X = x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338

8. Example of the Variance Add up the products to get the variance. Add up the products to get the variance. x P(X = x) x – µ (x – µ) 2 (x – µ) 2  P(X = x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338 0.810 =  2

9. Example of the Variance Add up the products to get the variance. Add up the products to get the variance. x P(X = x) x – µ (x – µ) 2 (x – µ) 2  P(X = x) 00.10-1.72.890.289 10.30-0.70.490.147 20.40+0.30.090.036 30.20+1.31.690.338 0.810 =  2 0.9 = 

10. Exercise Let’s Return To Let’s Do It! 7.23, p. 430. Let’s Return To Let’s Do It! 7.23, p. 430. The variance of Profit Indoors is The variance of Profit Indoors is Var(Profit Indoors) = 29.0 Compute the variance of Profit Outdoors. Compute the variance of Profit Outdoors. Which setting (indoors or outdoors) exhibits the greater variability? Which setting (indoors or outdoors) exhibits the greater variability?

11. Alternate Formula for the Variance It turns out that It turns out that Var(X) = E(X 2 ) – (E(X)) 2. That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X.” That is, the variance of X is “the expected value of the square of X minus the square of the expected value of X.” Of course, we could write this as Of course, we could write this as Var(X) = E(X 2 ) – µ 2.

12. Example of the Variance One more time, let X be the number of children in a household. One more time, let X be the number of children in a household. x P(X = x) 00.10 10.30 20.40 30.20

13. Example of the Variance Square each value of X. Square each value of X. x P(X = x) x2x2x2x2 00.100 10.301 20.404 30.209

14. Example of the Variance Multiply each squared X by its probability. Multiply each squared X by its probability. x P(X = x) x2x2x2x2 x 2  P(X = x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80

15. Example of the Variance Add up the products to get E(X 2 ). Add up the products to get E(X 2 ). x P(X = x) x2x2x2x2 x 2  P(X = x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80 3.70 = E(X 2 )

16. Example of the Variance Then use E(X 2 ) to compute the variance. Then use E(X 2 ) to compute the variance. Var(X) = E(X 2 ) – µ 2 Var(X) = E(X 2 ) – µ 2 = 3.70 – (1.7) 2 = 3.70 – 2.89 = 0.81. It follows that  =  0.81 = 0.9. It follows that  =  0.81 = 0.9.

17. Exercise Return once more to Let’s Do It! 7.23, p. 430. Return once more to Let’s Do It! 7.23, p. 430. Use the alternate formula to compute the variance of Profit Indoors. Use the alternate formula to compute the variance of Profit Indoors.

18. Means and Standard Deviations on the TI-83 Store the list of values of X in L 1. Store the list of values of X in L 1. Store the list of probabilities of X in L 2. Store the list of probabilities of X in L 2. Select STAT > CALC > 1-Var Stats. Select STAT > CALC > 1-Var Stats. Press ENTER. Press ENTER. Enter L 1, L 2. Enter L 1, L 2. Press ENTER. Press ENTER. The list of statistics includes the mean and standard deviation of X. The list of statistics includes the mean and standard deviation of X.

19. Means and Standard Deviations on the TI-83 Let L 1 = {0, 1, 2, 3}. Let L 1 = {0, 1, 2, 3}. Let L 2 = {0.1, 0.3, 0.4, 0.2}. Let L 2 = {0.1, 0.3, 0.4, 0.2}. Compute the statistics. Compute the statistics. Compute µ and  for the Indoor and Outdoor distributions in Let’s Do It! 7.23, p. 430. Compute µ and  for the Indoor and Outdoor distributions in Let’s Do It! 7.23, p. 430.

20. Assignment Page 442: Exercise 56. Page 442: Exercise 56. Page 451: Exercises 91 *, 93 *, 94 *, 95 *, 96 *. Page 451: Exercises 91 *, 93 *, 94 *, 95 *, 96 *. * Find the variance and standard deviation.