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

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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 Boole’s 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: . ] [ ] [ ] [ 2 2 Y E X E XY E Example There are three random variables X , Y , and Z with v a r ( ) v

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• Spring '08
• Tucker,A
• Probability theory, probability density function, Probability mass function, joint probability density, conditional variance formula

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

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