1. Mean and Standard Deviation Lecture 23 Section 7.5.1 Fri, Mar 3, 2006

2. The Mean and Standard Deviation Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run. Mean of a Discrete Random Variable – The average of the values that the random variable takes on, in the long run. Standard Deviation of a Discrete Random Variable – The standard deviation of the values that the random variable takes on, in the long run. Standard Deviation of a Discrete Random Variable – The standard deviation of the values that the random variable takes on, in the long run.

3. The Mean of a Discrete Random Variable The mean is also called the expected value. The mean is also called the expected value. If X is weight, then the mean of X is the expected weight, etc. If X is weight, then the mean of X is the expected weight, etc. However, that does not mean that it is the value that we literally expect to see. However, that does not mean that it is the value that we literally expect to see. “Expected value” is simply a synonym for the mean or average. “Expected value” is simply a synonym for the mean or average.

4. The Mean of a Discrete Random Variable The mean, or expected value, of X may be denoted by either of two symbols. The mean, or expected value, of X may be denoted by either of two symbols. µ or E(X) If another random variable is called Y, then we would write E(Y). If another random variable is called Y, then we would write E(Y). Or we could write them as µ X and µ Y. Or we could write them as µ X and µ Y.

5. Computing the Mean Given the pdf of X, the mean is computed as Given the pdf of X, the mean is computed as This is a weighted average of X. This is a weighted average of X. Each value is weighted by its likelihood. Each value is weighted by its likelihood.

6. Example of the Mean Recall the example where X was the number of children in a household. Recall the example where X was the number of children in a household. xP(x)P(x) 00.10 10.30 20.40 30.20

7. Example of the Mean xP(x)P(x) xP(x)xP(x) 00.100.00 10.300.30 20.400.80 30.200.60 Multiply each x by the corresponding probability. Multiply each x by the corresponding probability.

8. Example of the Mean xP(x)P(x) xP(x)xP(x) 00.100.00 10.300.30 20.400.80 30.200.60 1.70 = µ Add up the column of products to get the mean. Add up the column of products to get the mean.

9. Example: Powerball Use the handout to calculate the expected value of a Powerball ticket. Use the handout to calculate the expected value of a Powerball ticket. www.powerball.com www.powerball.com www.powerball.com

10. The Variance of a Discrete Random Variable Variance of a Discrete Random Variable – The average squared deviation of the values that the random variable takes on, in the long run. Variance of a Discrete Random Variable – The average squared deviation of the values that the random variable takes on, in the long run. 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 .

11. 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,

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

13. Example of the Variance xP(x)P(x)x – µ 00.10-1.7 10.30-0.7 20.40+0.3 30.20+1.3 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.

14. Example of the Variance xP(x)P(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 Square the deviations. Square the deviations.

15. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(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 Multiply each squared deviation by its probability. Multiply each squared deviation by its probability.

16. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(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 Add up the products to get the variance. Add up the products to get the variance.

17. Example of the Variance xP(x)P(x)x – µ(x – µ) 2 (x – µ) 2  P(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 =  Take the square root to get the standard deviation. Take the square root to get the standard deviation.

18. Alternate Formula for the Variance It turns out that It turns out that 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

19. Example of the Variance xP(x)P(x) 00.10 10.30 20.40 30.20 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.

20. Example of the Variance xP(x)P(x)x2x2 00.100 10.301 20.404 30.209 Square each value of X. Square each value of X.

21. Example of the Variance xP(x)P(x)x2x2 x2P(x)x2P(x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80 Multiply each squared X by its probability. Multiply each squared X by its probability.

22. Example of the Variance xP(x)P(x)x2x2 x2P(x)x2P(x) 00.1000.00 10.3010.30 20.4041.60 30.2091.80 3.70 = E(X 2 ) Add up the products to get E(X 2 ). Add up the products to get E(X 2 ).

23. Example of the Variance Then use E(X 2 ) and µ to compute the variance. Then use E(X 2 ) and µ 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.

24. TI-83 – Means and Standard Deviations 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. Use  x, not Sx, for the standard deviation. Use  x, not Sx, for the standard deviation.

25. TI-83 – Means and Standard Deviations 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 parameters  and . Compute the parameters  and .