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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/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/

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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 have a red die which, when thrown, takes the numbers from 1 to 6 with equal probability.

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

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

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

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red123456 green 1 2 3 4 57 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

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

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 7 X 2 3 4 5 6 7 8 9 10 11 12 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

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 8 Xf 2 3 4 5 6 7 8 9 10 11 12 We will now define f, the frequencies associated with the possible values of X. PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 9 Xf 2 3 4 54 6 7 8 9 10 11 12 For example, there are four outcomes which make X equal to 5. PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 10 Xf 21 32 43 54 65 76 85 94 103 112 121 Similarly you can work out the frequencies for all the other values of X. PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 11 Xfp 21 32 43 54 65 76 85 94 103 112 121 Finally we will derive the probability of obtaining each value of X. PROBABILITY DISTRIBUTION EXAMPLE: X IS THE SUM OF TWO DICE

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 12 Xfp 21 32 43 54 65 76 85 94 103 112 121 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

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red123456 green 1234567 2345678 3456789 45678910 567891011 6789101112 13 Xfp 211/36 322/36 433/36 544/36 655/36 766/36 855/36 944/36 1033/36 1122/36 1211/36 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

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

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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/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 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/lsewww.londoninternational.ac.uk/lse. 11.07.25

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Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:

Christopher Dougherty EC220 - Introduction to econometrics (review chapter) Slideshow: expected value of a function of a random variable Original citation:

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