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1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 This sequence provides an example of a discrete random variable. Suppose that you.

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Presentation on theme: "1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 This sequence provides an example of a discrete random variable. Suppose that you."— Presentation transcript:

1 1 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 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 2 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 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.

3 3 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 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.

4 4 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1 2 3 4 5 610 For example, if the red die is 4 and the green one is 6, X is equal to 10.

5 5 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE Similarly, if the red die is 2 and the green one is 5, X is equal to 7. red123456 green 1 2 3 4 57 6

6 6 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 The table shows all the possible outcomes.

7 7 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 If you look at the table, you can see that X can be any of the numbers from 2 to 12.

8 8 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 We will now define f, the frequencies associated with the possible values of X.

9 9 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 For example, there are four outcomes which make X equal to 5.

10 10 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Similarly you can work out the frequencies for all the other values of X. Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36

11 11 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 Finally we will derive the probability of obtaining each value of X.

12 12 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 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 13 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 red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36

14 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 23 456789101112X 1 36 1 36 PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

15 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/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 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/lsewww.londoninternational.ac.uk/lse. 2012.10.29

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