Christopher Dougherty

EC220 - Introduction to econometrics (review chapter)Slideshow: probability distribution example: x is the sum of two dice

 

 

 

 

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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1

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6

This sequence provides an example of a discrete random variable. Suppose that you have a red die which, when thrown, takes the numbers from 1 to 6 with equal probability.

2

red 1 2 3 4 5 6 green

1

2

3

4

5

6

Suppose that you also have a green die that can take the numbers from 1 to 6 with equal probability.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

3

red 1 2 3 4 5 6 green

1

2

3

4

5

6

We will define a random variable X as the sum of the numbers when the dice are thrown.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

4

For example, if the red die is 4 and the green one is 6, X is equal to 10.

red 1 2 3 4 5 6 green

1

2

3

4

5

6 10

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1

2

3

4

5 7

6

5

Similarly, if the red die is 2 and the green one is 5, X is equal to 7.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

6

The table shows all the possible outcomes.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

7

X 23456789

101112

If you look at the table, you can see that X can be any of the numbers from 2 to 12.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

8

X f 23456789

101112

We will now define f, the frequencies associated with the possible values of X.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

9

X f 2345 46789

101112

For example, there are four outcomes which make X equal to 5.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

10

X f 2 13 24 35 46 57 68 59 4

10 311 212 1

Similarly you can work out the frequencies for all the other values of X.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

11

X f p 2 13 24 35 46 57 68 59 4

10 311 212 1

Finally we will derive the probability of obtaining each value of X.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

12

X f p 2 13 24 35 46 57 68 59 4

10 311 212 1

If there is 1/6 probability of obtaining each number on the red die, and the same on the green die, each outcome in the table will occur with 1/36 probability.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

red 1 2 3 4 5 6 green

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 7 8 9

4 5 6 7 8 9 10

5 6 7 8 9 10 11

6 7 8 9 10 11 12

13

X f p 2 1 1/363 2 2/364 3 3/365 4 4/366 5 5/367 6 6/368 5 5/369 4 4/36

10 3 3/3611 2 2/3612 1 1/36

Hence to obtain the probabilities associated with the different values of X, we divide the frequencies by 36.

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

14

The distribution is shown graphically. in this example it is symmetrical, highest for X equal to 7 and declining on either side.

6

__

36

5

__

36

4

__

36

3

__

36

2

__

36

2

__

36

3

__

36

5

__

36

4

__

36

probability

2 3 4 5 6 7 8 9 10 11 12 X

PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

1

36

1

36

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