Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a random variable Original citation: Dougherty, C. (2012) EC220 - Introduction to econometrics (review chapter). [Teaching Resource] © 2012 The Author This version available at: Available in LSE Learning Resources Online: May 2012 This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 License. This license allows the user to remix, tweak, and build upon the work even for commercial purposes, as long as the user credits the author and licenses their new creations under the identical terms

EXPECTED VALUE OF A RANDOM VARIABLE 1 The expected value of a random variable, also known as its population mean, is the weighted average of its possible values, the weights being the probabilities attached to the values. Definition of E(X), the expected value of X :

EXPECTED VALUE OF A RANDOM VARIABLE 2 Note that the sum of the probabilities must be unity, so there is no need to divide by the sum of the weights.

x i x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 3 EXPECTED VALUE OF A RANDOM VARIABLE This sequence shows how the expected value is calculated, first in abstract and then with the random variable defined in the first sequence. We begin by listing the possible values of X.

x i p i x 1 p 1 x 2 p 2 x 3 p 3 x 4 p 4 x 5 p 5 x 6 p 6 x 7 p 7 x 8 p 8 x 9 p 9 x 10 p 10 x 11 p 11 4 EXPECTED VALUE OF A RANDOM VARIABLE Next we list the probabilities attached to the different possible values of X.

x i p i x 1 p 1 x 2 p 2 x 3 p 3 x 4 p 4 x 5 p 5 x 6 p 6 x 7 p 7 x 8 p 8 x 9 p 9 x 10 p 10 x 11 p 11 5 EXPECTED VALUE OF A RANDOM VARIABLE Then we define a column in which the values are weighted by the corresponding probabilities.

x i p i x 1 p 1 x 2 p 2 x 3 p 3 x 4 p 4 x 5 p 5 x 6 p 6 x 7 p 7 x 8 p 8 x 9 p 9 x 10 p 10 x 11 p 11 6 EXPECTED VALUE OF A RANDOM VARIABLE We do this for each value separately.

x i p i x 1 p 1 x 2 p 2 x 3 p 3 x 4 p 4 x 5 p 5 x 6 p 6 x 7 p 7 x 8 p 8 x 9 p 9 x 10 p 10 x 11 p 11 7 EXPECTED VALUE OF A RANDOM VARIABLE Here we are assuming that n, the number of possible values, is equal to 11, but it could be any number.

x i p i x 1 p 1 x 2 p 2 x 3 p 3 x 4 p 4 x 5 p 5 x 6 p 6 x 7 p 7 x 8 p 8 x 9 p 9 x 10 p 10 x 11 p 11  x i p i = E(X) 8 EXPECTED VALUE OF A RANDOM VARIABLE The expected value is the sum of the entries in the third column.

x i p i x i p i x i p i x 1 p 1 x 1 p 1 21/36 x 2 p 2 x 2 p 2 32/36 x 3 p 3 x 3 p 3 43/36 x 4 p 4 x 4 p 4 54/36 x 5 p 5 x 5 p 5 65/36 x 6 p 6 x 6 p 6 76/36 x 7 p 7 x 7 p 7 85/36 x 8 p 8 x 8 p 8 94/36 x 9 p 9 x 9 p 9 103/36 x 10 p 10 x 10 p /36 x 11 p 11 x 11 p /36  x i p i = E(X) 9 EXPECTED VALUE OF A RANDOM VARIABLE The random variable X defined in the previous sequence could be any of the integers from 2 to 12 with probabilities as shown.

x i p i x i p i x 1 p 1 x 1 p 1 21/362/36 x 2 p 2 x 2 p 2 32/36 x 3 p 3 x 3 p 3 43/36 x 4 p 4 x 4 p 4 54/36 x 5 p 5 x 5 p 5 65/36 x 6 p 6 x 6 p 6 76/36 x 7 p 7 x 7 p 7 85/36 x 8 p 8 x 8 p 8 94/36 x 9 p 9 x 9 p 9 103/36 x 10 p 10 x 10 p /36 x 11 p 11 x 11 p /36  x i p i = E(X) 10 EXPECTED VALUE OF A RANDOM VARIABLE X could be equal to 2 with probability 1/36, so the first entry in the calculation of the expected value is 2/36.

x i p i x i p i x 1 p 1 x 1 p 1 21/362/36 x 2 p 2 x 2 p 2 32/366/36 x 3 p 3 x 3 p 3 43/36 x 4 p 4 x 4 p 4 54/36 x 5 p 5 x 5 p 5 65/36 x 6 p 6 x 6 p 6 76/36 x 7 p 7 x 7 p 7 85/36 x 8 p 8 x 8 p 8 94/36 x 9 p 9 x 9 p 9 103/36 x 10 p 10 x 10 p /36 x 11 p 11 x 11 p /36  x i p i = E(X) 11 EXPECTED VALUE OF A RANDOM VARIABLE The probability of x being equal to 3 was 2/36, so the second entry is 6/36.

x i p i x i p i x 1 p 1 x 1 p 1 21/362/36 x 2 p 2 x 2 p 2 32/366/36 x 3 p 3 x 3 p 3 43/3612/36 x 4 p 4 x 4 p 4 54/3620/36 x 5 p 5 x 5 p 5 65/3630/36 x 6 p 6 x 6 p 6 76/3642/36 x 7 p 7 x 7 p 7 85/3640/36 x 8 p 8 x 8 p 8 94/3636/36 x 9 p 9 x 9 p 9 103/3630/36 x 10 p 10 x 10 p /3622/36 x 11 p 11 x 11 p /3612/36  x i p i = E(X) 12 EXPECTED VALUE OF A RANDOM VARIABLE Similarly for the other 9 possible values.

x i p i x i p i x 1 p 1 x 1 p 1 21/362/36 x 2 p 2 x 2 p 2 32/366/36 x 3 p 3 x 3 p 3 43/3612/36 x 4 p 4 x 4 p 4 54/3620/36 x 5 p 5 x 5 p 5 65/3630/36 x 6 p 6 x 6 p 6 76/3642/36 x 7 p 7 x 7 p 7 85/3640/36 x 8 p 8 x 8 p 8 94/3636/36 x 9 p 9 x 9 p 9 103/3630/36 x 10 p 10 x 10 p /3622/36 x 11 p 11 x 11 p /3612/36  x i p i = E(X) 252/36 13 To obtain the expected value, we sum the entries in this column. EXPECTED VALUE OF A RANDOM VARIABLE

14 The expected value turns out to be 7. Actually, this was obvious anyway. We saw in the previous sequence that the distribution is symmetrical about 7. EXPECTED VALUE OF A RANDOM VARIABLE x i p i x i p i x 1 p 1 x 1 p 1 21/362/36 x 2 p 2 x 2 p 2 32/366/36 x 3 p 3 x 3 p 3 43/3612/36 x 4 p 4 x 4 p 4 54/3620/36 x 5 p 5 x 5 p 5 65/3630/36 x 6 p 6 x 6 p 6 76/3642/36 x 7 p 7 x 7 p 7 85/3640/36 x 8 p 8 x 8 p 8 94/3636/36 x 9 p 9 x 9 p 9 103/3630/36 x 10 p 10 x 10 p /3622/36 x 11 p 11 x 11 p /3612/36  x i p i = E(X) 252/36 = 7

Very often the expected value of a random variable is represented by , the Greek m. If there is more than one random variable, their expected values are differentiated by adding subscripts to . 15 EXPECTED VALUE OF A RANDOM VARIABLE Definition of E(X), the expected value of X : Alternative notation for E(X) : E(X) =  X

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