x k withmeans 1 2 k theexpectedvalueoftheirsumis 29

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Unformatted text preview: 5 Let g(X, Y) be a function of the discrete random variables X and Y. The expected value of this function is defined as: E[g(X , Y )] = ∑ ∑ g(x , y ) PX , Y (x , y ) xy A property of expectation is: E(X + Y ) = E(X ) + E(Y ) This result can be shown: E(X + Y ) = ∑ ∑ (x + y ) PX , Y (x , y ) xy [ = ∑ ∑ x PX , Y (x , y ) + y PX , Y (x , y ) xy = ∑ x ∑ PX , Y (x , y ) + ∑ y ∑ PX , Y (x , y ) x y y x = ∑ x PX (x) + ∑ y PY (y ) x y = E(X ) + E(Y ) For constant fixed numbers a and b a rule is: E(a X + b Y ) = a E(X ) + b E(Y ) A general result is that for K random variables X 1 , X 2 , . . . , X K with means μ 1 , μ 2 , . . . , μ K the expected value of their sum is: 29 E(X 1 + X 2 + K + X K ) = μ 1 + μ 2 + K + μ K Chapter 5 A measure of a linear relationship between two random variables is of interest. For random variables X and Y with means μ X and μ Y the covariance between X and Y is defined as: Cov(X , Y ) = E[(X − μ X ) (Y − μ Y )] = ∑ ∑ (x − μ X ) (y − μ Y ) PX , Y (x , y ) xy An equiva...
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This note was uploaded on 02/06/2014 for the course ECON ECON 325 taught by Professor Whistler during the Spring '10 term at The University of British Columbia.

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