Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected...

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: http://learningresources.lse.ac.uk/141/

Available in LSE Learning Resources Online: May 2012

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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:

n

iiinn pxpxpxXE

111 ...)(



2

Note that the sum of the probabilities must be unity, so there is no need to divide by the sum of the weights.

n

iiinn pxpxpxXE

111 ...)(

xi

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

3


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.

xi pi

x1 p1

x2 p2

x3 p3

x4 p4

x5 p5

x6 p6

x7 p7

x8 p8

x9 p9

x10 p10

x11 p11

4


Next we list the probabilities attached to the different possible values of X.

xi pi xi pi

x1 p1 x1 p1

x2 p2

x3 p3

x4 p4

x5 p5

x6 p6

x7 p7

x8 p8

x9 p9

x10 p10

x11 p11

5


Then we define a column in which the values are weighted by the corresponding probabilities.

xi pi xi pi

x1 p1 x1 p1

x2 p2 x2 p2

x3 p3

x4 p4

x5 p5

x6 p6

x7 p7

x8 p8

x9 p9

x10 p10

x11 p11

6


We do this for each value separately.

xi pi xi pi

x1 p1 x1 p1

x2 p2 x2 p2

x3 p3 x3 p3

x4 p4 x4 p4

x5 p5 x5 p5

x6 p6 x6 p6

x7 p7 x7 p7

x8 p8 x8 p8

x9 p9 x9 p9

x10 p10 x10 p10

x11 p11 x11 p11

7


Here we are assuming that n, the number of possible values, is equal to 11, but it could be any number.

xi pi xi pi

x1 p1 x1 p1

x2 p2 x2 p2

x3 p3 x3 p3

x4 p4 x4 p4

x5 p5 x5 p5

x6 p6 x6 p6

x7 p7 x7 p7

x8 p8 x8 p8

x9 p9 x9 p9

x10 p10 x10 p10

x11 p11 x11 p11

xi pi = E(X)

8


The expected value is the sum of the entries in the third column.

xi pi xi pi xi pi

x1 p1 x1 p1 2 1/36

x2 p2 x2 p2 3 2/36

x3 p3 x3 p3 4 3/36

x4 p4 x4 p4 5 4/36

x5 p5 x5 p5 6 5/36

x6 p6 x6 p6 7 6/36

x7 p7 x7 p7 8 5/36

x8 p8 x8 p8 9 4/36

x9 p9 x9 p9 10 3/36

x10 p10 x10 p10 11 2/36

x11 p11 x11 p11 12 1/36

xi pi = E(X)

9


The random variable X defined in the previous sequence could be any of the integers from 2 to 12 with probabilities as shown.

xi pi xi pi xi pi xi pi

x1 p1 x1 p1 2 1/36 2/36

x2 p2 x2 p2 3 2/36

x3 p3 x3 p3 4 3/36

x4 p4 x4 p4 5 4/36

x5 p5 x5 p5 6 5/36

x6 p6 x6 p6 7 6/36

x7 p7 x7 p7 8 5/36

x8 p8 x8 p8 9 4/36

x9 p9 x9 p9 10 3/36

x10 p10 x10 p10 11 2/36

x11 p11 x11 p11 12 1/36

xi pi = E(X)

10


X could be equal to 2 with probability 1/36, so the first entry in the calculation of the expected value is 2/36.


x1 p1 x1 p1 2 1/36 2/36

x2 p2 x2 p2 3 2/36 6/36

x3 p3 x3 p3 4 3/36

x4 p4 x4 p4 5 4/36

x5 p5 x5 p5 6 5/36

x6 p6 x6 p6 7 6/36

x7 p7 x7 p7 8 5/36

x8 p8 x8 p8 9 4/36

x9 p9 x9 p9 10 3/36

x10 p10 x10 p10 11 2/36

x11 p11 x11 p11 12 1/36

xi pi = E(X)

11


The probability of x being equal to 3 was 2/36, so the second entry is 6/36.


x1 p1 x1 p1 2 1/36 2/36

x2 p2 x2 p2 3 2/36 6/36

x3 p3 x3 p3 4 3/36 12/36

x4 p4 x4 p4 5 4/36 20/36

x5 p5 x5 p5 6 5/36 30/36

x6 p6 x6 p6 7 6/36 42/36

x7 p7 x7 p7 8 5/36 40/36

x8 p8 x8 p8 9 4/36 36/36

x9 p9 x9 p9 10 3/36 30/36

x10 p10 x10 p10 11 2/36 22/36

x11 p11 x11 p11 12 1/36 12/36

xi pi = E(X)

12


Similarly for the other 9 possible values.


x1 p1 x1 p1 2 1/36 2/36

x2 p2 x2 p2 3 2/36 6/36

x3 p3 x3 p3 4 3/36 12/36

x4 p4 x4 p4 5 4/36 20/36

x5 p5 x5 p5 6 5/36 30/36

x6 p6 x6 p6 7 6/36 42/36

x7 p7 x7 p7 8 5/36 40/36

x8 p8 x8 p8 9 4/36 36/36

x9 p9 x9 p9 10 3/36 30/36

x10 p10 x10 p10 11 2/36 22/36

x11 p11 x11 p11 12 1/36 12/36

xi pi = E(X) 252/36

13

To obtain the expected value, we sum the entries in this column.


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.



x1 p1 x1 p1 2 1/36 2/36

x2 p2 x2 p2 3 2/36 6/36

x3 p3 x3 p3 4 3/36 12/36

x4 p4 x4 p4 5 4/36 20/36

x5 p5 x5 p5 6 5/36 30/36

x6 p6 x6 p6 7 6/36 42/36

x7 p7 x7 p7 8 5/36 40/36

x8 p8 x8 p8 9 4/36 36/36

x9 p9 x9 p9 10 3/36 30/36

x10 p10 x10 p10 11 2/36 22/36

x11 p11 x11 p11 12 1/36 12/36

xi pi = 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



Alternative notation for E(X): E(X) = X

n

iiinn pxpxpxXE

111 ...)(

Copyright Christopher Dougherty 2011.

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/.

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

http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx

or the University of London International Programmes distance learning course

20 Elements of Econometrics

www.londoninternational.ac.uk/lse.

11.07.25

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