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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 i i i n n p x p x p x X E 1 1 1 ... ) (

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Page 1: 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

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 ...)(

Page 2: 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

2

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

EXPECTED VALUE OF A RANDOM VARIABLE

Definition of E(X), the expected value of X:

n

iiinn pxpxpxXE

111 ...)(

Page 3: 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

3

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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) 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.

Page 4: 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

4

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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

Page 5: 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

5

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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

Page 6: 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

6

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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) We do this for each value separately.

Page 7: 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

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.

xi pi xi pi xi pi xi pi

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)

Page 8: 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

8

EXPECTED VALUE OF A RANDOM VARIABLE

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

xi pi xi pi xi pi xi pi

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)

Page 9: 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

9

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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

Page 10: 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

10

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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

Page 11: 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

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.

xi pi xi pi xi pi xi pi

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)

Page 12: 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

12

EXPECTED VALUE OF A RANDOM VARIABLE

xi pi xi pi xi pi xi pi

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) Similarly for the other 9 possible values.

Page 13: 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

xi pi xi pi xi pi xi pi

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.

EXPECTED VALUE OF A RANDOM VARIABLE

Page 14: 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

xi pi xi pi xi pi xi pi

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

14

EXPECTED VALUE OF A RANDOM VARIABLE

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.

Page 15: 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

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

Alternative notation for E(X):

X

n

iiinn pxpxpxXE

111 ...)(

Page 16: 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

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/.

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

or the University of London International Programmes distance learning course

EC2020 Elements of Econometrics

www.londoninternational.ac.uk/lse.

2012.10.29


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