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1 This very short sequence presents an important definition, that of the independence of two random variables. Two random variables X and Y are said to be independent if and only if E[f(X)g(Y)] = E[f(X)] E[g(Y)] for any functions f(X) and g(Y). INDEPENDENCE OF TWO RANDOM VARIABLES

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2 Two variables X and Y are independent if and only if, given any functions f(X) and g(Y), the expected value of the product f(X)g(Y) is equal to the expected value of f(X) multiplied by the expected value of g(Y). Two random variables X and Y are said to be independent if and only if E[f(X)g(Y)] = E[f(X)] E[g(Y)] for any functions f(X) and g(Y). INDEPENDENCE OF TWO RANDOM VARIABLES

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Two random variables X and Y are said to be independent if and only if E[f(X)g(Y)] = E[f(X)] E[g(Y)] for any functions f(X) and g(Y). Special case: if X and Y are independent, E(X Y) = E(X) E(Y) 3 INDEPENDENCE OF TWO RANDOM VARIABLES As a special case, the expected value of XY is equal to the expected value of X multiplied by the expected value of Y if and only if X and Y are independent.

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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.3 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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THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components.

THE FIXED AND RANDOM COMPONENTS OF A RANDOM VARIABLE 1 In this short sequence we shall decompose a random variable X into its fixed and random components.

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