1. The third sequence defined the expected value of a function of a random variable X. There is only one function that is of much interest to us, at least initially: the squared deviation from the population mean. 1 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Population variance of X:

2. The expected value of the squared deviation is known as the population variance of X. It is a measure of the dispersion of the distribution of X about its population mean. 2 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Population variance of X:

3. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 3 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE We will calculate the population variance of the random variable X defined in the first sequence. We start as usual by listing the possible values of X and the corresponding probabilities.

4. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 4 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Next we need a column giving the deviations of the possible values of X about its population mean. In the second sequence we saw that the population mean of X was 7.

5. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 5 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE When X is equal to 2, the deviation is –5.

6. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 6 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Similarly for all the other possible values.

7. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 7 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Next we need a column giving the squared deviations. When X is equal to 2, the squared deviation is 25.

8. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 8 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Similarly for the other values of X.

9. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 9 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Now we start weighting the squared deviations by the corresponding probabilities. What do you think the weighted average will be? Have a guess.

10. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 10 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE A reason for making an initial guess is that it may help you to identify an arithmetical error, if you make one. If the initial guess and the outcome are very different, that is a warning.

11. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 11 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE We calculate all the weighted squared deviations.

12. x i p i x i –  (x i –  ) 2 (x i –  ) 2 p i 21/36–5250.69 32/36–4160.89 43/36–390.75 54/36–240.44 65/36–110.14 76/36000.00 85/36110.14 94/36240.44 103/36390.75 112/364160.89 121/365250.69 5.83 The sum is the population variance of X. 12 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE

13. Population variance of X In equations, the population variance of X is usually written  X 2,  being the Greek s. 13 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE

14. The standard deviation of X is the square root of its population variance. Usually written  x, it is an alternative measure of dispersion. It has the same units as X. 14 POPULATION VARIANCE OF A DISCRETE RANDOM VARIABLE Standard deviation of X

15. Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author. The content of this slideshow comes from Section R.2 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course EC212 Introduction to Econometrics http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course EC2020 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.10.29