L16_S10 - AMS 311, Spring Semester, 2010 Chapter Seven...

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Unformatted text preview: AMS 311, Spring Semester, 2010 Chapter Seven Properties of Expectation Proposition 2.1 If X and Y have a joint probability mass function p(x,y) , then , ) , ( ) , ( )] , ( [ = y x y x p y x g Y X g E provided that the sum is absolutely convergent. If X and Y have a joint probability density function f(x,y) , then , ) , ( ) , ( )] , ( [ - - = dxdy y x f y x g Y X g E provided that the integral is absolutely convergent. A corollary of this theorem 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. In Example 2d, this result is used to study Booles inequality. Example 2g . Mean of a hypergeometric random variable . If n balls are randomly selected from an urn containing N balls of which m are white, find the expected number of white balls selected. Answer is . N mn Example 2q . Let X be a nonnegative, integer-valued random variable. Prove that . } [ ] [ 1 = = n i i X P X E Cauchy-Schwartz inequality generalizes to expectations: . ] [ ] [ ]...
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This note was uploaded on 04/13/2010 for the course AMS 311 taught by Professor Tucker,a during the Spring '08 term at SUNY Stony Brook.

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L16_S10 - AMS 311, Spring Semester, 2010 Chapter Seven...

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