1. 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/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://creativecommons.org/licenses/by-sa/3.0/ http://learningresources.lse.ac.uk/

2. 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.

3. 2 red123456 green 1 2 3 4 5 6 Suppose that one die is red and the other green. EXERCISE R.2

4. 3 red123456 green 1 2 3 4 56 Then, for example, if the red die is 4 and the green one is 6, X is equal to 6.

5. 4 EXERCISE R.2 red123456 green 1 2 3 45 6 Similarly, if the red die is 2 and the green one is 5, X is equal to 5.

6. 5 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 The table shows all the possible outcomes.

7. 6 X123456X123456 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 If you look at the table, you can see that X can be any of the numbers from 1 to 6.

8. 7 Xf123456Xf123456 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 We will now define f, the frequencies associated with the possible values of X.

9. 8 Xf1234756Xf1234756 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 For example, there are seven outcomes which make X equal to 4.

10. 9 Xf1 23 35 47 59 611 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 Similarly you can work out the frequencies for the other values of X.

11. 10 Xfp1 23 35 47 59 611 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 Finally we will derive the probability of obtaining each value of X.

12. 11 Xfp1 23 35 47 59 611 EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666 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.

13. 12 Xfp 111/36 233/36 355/36 477/36 599/36 61111/36 Hence to obtain the probabilities associated with the different values of X, we divide the frequencies by 36. EXERCISE R.2 red123456 green 1123456 2223456 3333456 4444456 5555556 6666666

14. Copyright Christopher Dougherty 1999–2006. This slideshow may be freely copied for personal use. 26.08.06