week1 - Econ 508: Discussion 28 Jan. 2011 Juan Fung 1...

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28 Jan. 2011 Juan Fung 1 Review: Functions of discrete random variables Let’s go over some basic properties of the expectations operator. Suppose X takes values in { X 1 ,...,X N } , with respective probabilities { p 1 ,...,p N } , i p i = 1. Then the expected value of X is E [ X ] = N X i =1 p i X i , and the variance of X is V ar [ X ] = E ( X - E [ X ]) 2 = N X i =1 p i ( X i - E [ X ]) 2 . Let Y be another discrete random variable. Then the covariance between X and Y is Cov [ X,Y ] = E ( X - E [ X ])( Y - E [ Y ]) . 1. Let a k R be constants. Use your thorough knowledge from Econ 506 to find the following: (a) E [( a 1 X ) 2 - a 2 1 Y + a 3 ] = a 2 1 E [ X 2 ] - a 2 E [ 1 Y ] + a 3 . (b) V ar [( a 1 X ) 2 - a 2 1 Y + a 3 ] = a 4 1 V ar [ X 2 ] + a 2 2 V ar [ 1 Y ] - 2 a 2 1 a 2 Cov [ X 2 , 1 Y ]. Lesson : be careful with nonlinear transformations! 2. Consider the joint probability distribution of X and Y given below: X 5 10 15 20 25 Y 10 8 10 15 12 Prob 1/5 1/5
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week1 - Econ 508: Discussion 28 Jan. 2011 Juan Fung 1...

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