Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: exercise r.2 Original citation: Dougherty, C. (2012) EC220 - Introduction.

Christopher Dougherty

EC220 - Introduction to econometrics (review chapter)Slideshow: exercise r.2

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

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. http://creativecommons.org/licenses/by-sa/3.0/

 http://learningresources.lse.ac.uk/

1

EXERCISE R.2

R.2* A random variable X is defined to be the larger of the numbers when two dice are thrown, or the number if they are the same. Find the probability distribution for X.

2

red 1 2 3 4 5 6 green

1

2

3

4

5

6

Suppose that one die is red and the other green.

EXERCISE R.2

3

EXERCISE R.2

red 1 2 3 4 5 6 green

1

2

3

4

5

6 6

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

4

EXERCISE R.2

red 1 2 3 4 5 6 green

1

2

3

4

5 5

6

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

5

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

The table shows all the possible outcomes.

6

X 123456

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

If you look at the table, you can see that X can be any of the numbers from 1 to 6.

7

X f 123456

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

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

8

X f 1234 756

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

For example, there are seven outcomes which make X equal to 4.

9

X f 1 12 33 54 75 96 11

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

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

10

X f p 1 12 33 54 75 96 11

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

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

11

X f p 1 12 33 54 75 96 11

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

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.

12

X f p 1 1 1/362 3 3/363 5 5/364 7 7/365 9 9/366 11 11/36

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

EXERCISE R.2

red 1 2 3 4 5 6 green

1 1 2 3 4 5 6

2 2 2 3 4 5 6

3 3 3 3 4 5 6

4 4 4 4 4 5 6

5 5 5 5 5 5 6

6 6 6 6 6 6 6

Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use.

26.08.06