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1 EXPECTED VALUE RULES This sequence states the rules for manipulating expected values. First, the additive rule. The expected value of the sum of two random variables is the sum of their expected values. 1.

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2 EXPECTED VALUE RULES Here the sum consists of three variables. But the rule generalizes to any number. 1.

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3 EXPECTED VALUE RULES The second rule is the multiplicative rule. The expected value of a variable that has been multiplied by a constant) is equal to the constant multiplied by the expected value of the variable. 1. 2.

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4 EXPECTED VALUE RULES For example, the expected value of 3X is three times the expected value of X. Example: 1. 2.

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5 EXPECTED VALUE RULES Finally, the expected value of a constant is just the constant. Of course this is obvious. 1. 2. 3.

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1. 2. 3. 6 EXPECTED VALUE RULES As an exercise, we will use the rules to simplify the expected value of an expression. Suppose that we are interested in the expected value of a variable Y, where Y = b 1 + b 2 X.

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1. 2. 3. 7 EXPECTED VALUE RULES We use the first rule to break up the expected value into its two components.

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Then we use the second rule to replace E(b 2 X) by b 2 E(X) and the third rule to simplify E(b 1 ) to just b 1. This is as far as we can go in this example. 1. 8 EXPECTED VALUE RULES 2. 3.

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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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