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L15S10 - AMS 311 Spring Semester 2010 Chapter Seven...

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AMS 311, Spring Semester, 2010 Chapter Seven Properties of Expectation A corollary of Proposition 2.1 is that ], [ ] [ ] [ Y E X E Y X E + = + provided that the expectations are finite. By induction, ], [ ] [ ] [ ] [ 2 1 2 1 n n X E X E X E X X X E + + + = + + provided each expectation is finite. Example 2c . Let n X X X , , , 2 1 be jointly distributed random variables, each with expected value . μ Prove that . ] [ ] [ 1 μ = = = n X E X E n i i n Note that the only assumption needed is that the random variables have the same finite mean. Definition The covariance between X and Y , denoted by Cov ( X , Y ), is defined by ])]. [ ])( [ [( ) , ( Y E Y X E X E Y X Cov - - = One identity that is often helpful is that ]. [ ] [ ] [ ) , ( Y E X E XY E Y X Cov - = Proposition 3.2 (i) ). , cov( ) , cov( X Y Y X = (ii) ). var( ) , cov( X X X = (iii) ). , cov( ) , cov( X Y a Y aX = (iv) . ) , cov( ) , cov( 1 1 1 1 ∑∑ = = = = = n i m j j i m j j n i i Y X Y X Example 3a . Let n X X X , , , 2 1 be independent and identically distributed random variables having expected value μ and variance . 2 σ Let , 1 n X X n i i n = = and let . 1 ) ( 1 2 2 - - = = n X X S n i n i Compute ) var( n X and ]. [ 2 S E Definition The correlation of two random variables X and Y , denoted by ) , ( Y X ρ

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