To find the expected value of a function of a random variable, you calculate all the possible values of the function, weight them by the corresponding probabilities, and sum the results. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE Definition of E[g(X)], the expected value of a function of X : 1

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. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 2

xipix1p1x2p2x3p3……………………………………xnpn xipix1p1x2p2x3p3……………………………………xnpn ………………… The calculation of the expected value of a function of a random variable will be outlined in general and then illustrated with an example. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 3

xipix1p1x2p2x3p3……………………………………xnpn xipix1p1x2p2x3p3……………………………………xnpn ………………… First you list the possible values of X and the corresponding probabilities. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 4

x i p i g(x i ) x 1 p 1 g(x 1 ) x 2 p 2 g(x 2 ) x 3 p 3 g(x 3 ) ………... x n p n g(x n ) Next you calculate the function of X for each possible value of X. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 5

x i p i g(x i ) g(x i ) p i x 1 p 1 g(x 1 )g(x 1 ) p 1 x 2 p 2 g(x 2 ) x 3 p 3 g(x 3 ) ………... x n p n g(x n ) Then, one at a time, you weight the value of the function by its corresponding probability. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 6

x i p i g(x i ) g(x i ) p i x 1 p 1 g(x 1 )g(x 1 ) p 1 x 2 p 2 g(x 2 ) g(x 2 ) p 2 x 3 p 3 g(x 3 ) g(x 3 ) p 3 ………...……... x n p n g(x n ) g(x n ) p n You do this individually for each possible value of X. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 7

x i p i g(x i ) g(x i ) p i x 1 p 1 g(x 1 )g(x 1 ) p 1 x 2 p 2 g(x 2 ) g(x 2 ) p 2 x 3 p 3 g(x 3 ) g(x 3 ) p 3 ………...……... x n p n g(x n ) g(x n ) p n   g(x i ) p i The sum of the weighted values is the expected value of the function of X. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 8

x i p i g(x i ) g(x i ) p i x i p i x 1 p 1 g(x 1 )g(x 1 ) p 1 21/36 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/36 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/36 ………...……... 54/36 ………...……... 65/36 ………...……... 76/36 ………...……... 85/36 ………...……... 94/36 ………...…… /36 ………...…… /36 x n p n g(x n ) g(x n ) p n 121/36   g(x i ) p i 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. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 9

x i p i g(x i ) g(x i ) p i x i p i x i 2 x 1 p 1 g(x 1 )g(x 1 ) p 1 21/364 x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/369 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/3616 ………...……... 54/3625 ………...……... 65/3636 ………...……... 76/3649 ………...……... 85/3664 ………...……... 94/3681 ………...…… /36100 ………...…… /36121 x n p n g(x n ) g(x n ) p n 121/36144   g(x i ) p i First you calculate the possible values of X 2. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 10

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/ x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/369 x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/3616 ………...……... 54/3625 ………...……... 65/3636 ………...……... 76/3649 ………...……... 85/3664 ………...……... 94/3681 ………...…… /36100 ………...…… /36121 x n p n g(x n ) g(x n ) p n 121/36144   g(x i ) p i 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 EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 11

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/ x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/ x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/ ………...……... 54/ ………...……... 65/ ………...……... 76/ ………...……... 85/ ………...……... 94/ ………...…… / ………...…… / x n p n g(x n ) g(x n ) p n 121/   g(x i ) p i Similarly for all the other possible values of X. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 12

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/ x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/ x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/ ………...……... 54/ ………...……... 65/ ………...……... 76/ ………...……... 85/ ………...……... 94/ ………...…… / ………...…… / x n p n g(x n ) g(x n ) p n 121/   g(x i ) p i The expected value of X 2 is the sum of its weighted values in the final column. It is equal to It is the average value of the figures in the previous column, taking the differing probabilities into account. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 13

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/ x 2 p 2 g(x 2 ) g(x 2 ) p 2 32/ x 3 p 3 g(x 3 ) g(x 3 ) p 3 43/ ………...……... 54/ ………...……... 65/ ………...……... 76/ ………...……... 85/ ………...……... 94/ ………...…… / ………...…… / x n p n g(x n ) g(x n ) p n 121/   g(x i ) p i 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. EXPECTED VALUE OF A FUNCTION OF A RANDOM VARIABLE 14

Copyright Christopher Dougherty 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 Individuals studying econometrics on their own and 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 or the University of London International Programmes distance learning course 20 Elements of Econometrics