1. Definition of, the expected value of a function of X : 1 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE To find the expected value of a function of a random variable, one calculates all the possible values of the function, weights them by the corresponding probabilities, and sums the results.

2. Definition of, the expected value of a function of X : Example: For example, the expected value of X 2 is found by calculating all its possible values, multiplying them by the corresponding probabilities, and summing. 2 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE

3. 3 x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The calculation of the expected value of a function of a random variable will be outlined in general and then illustrated with an example.

4. 4 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 First one makes a list of the possible values of X and the corresponding probabilities.

5. 5 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Next the function of X is calculated for each possible value of X.

6. 6 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Then, one at a time, the value of the function is weighted by its corresponding probability.

7. 7 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 This is done individually for each possible value of X.

8. 8 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The sum of the weighted values is the expected value of the function of X.

9. 9 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The process will be illustrated for X 2, where X is the random variable defined in the first sequence. The 11 possible values of X and the corresponding probabilities are listed.

10. 10 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 First one calculates the possible values of X 2.

11. 11 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 The first value is 4, which arises when X is equal to 2. The probability of X being equal to 2 is 1/36, so the weighted function is 4/36, which we shall write in decimal form as 0.11.

12. 12 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Similarly for all the other possible values of X.

13. 13 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE The expected value of X 2 is the sum of its weighted values in the final column. It is equal to 54.83. It is the average value of the figures in the previous column, taking the differing probabilities into account. x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83

14. x i p i g(x i ) g(x i ) p i x i p i x i 2 x i 2 p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/3640.11 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/3690.50 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36161.33 ………...……... 54/36252.78 ………...……... 65/36365.00 ………...……... 76/36498.17 ………...……... 85/36648.89 ………...……... 94/36819.00 ………...……... 103/361008.83 ………...……... 112/361216.72 x n p n g(x n ) g(x n ) p n 121/361444.00   g(x i ) p i 54.83 Note that E(X 2 ) is not the same thing as E(X), squared. In the previous sequence we saw that E(X) for this example was 7. Its square is 49. 14 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE

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