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lecture7 - ISYE 2028 A and B Lecture 7 Conditional...

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ISYE 2028 A and B Lecture 7 Conditional Expectation and Prediction Dr. Kobi Abayomi February 10, 2009 1 Conditional Expectation (again) We should recall the definition of the conditional expectation: E ( X | Y = y ) = x x P ( x | Y = y ) , discrete R xf X | Y = y dx, continuous (1) In the case that (either) p y , f y are functions of only y (i.e. when Y 6 = g ( X ): E ( X | Y = y ) = ( P x x P ( x,y ) p y , discrete R xf X | Y = y dx f Y , continuous (2) If you are in doubt — i.e. you are trying to solve a problem — calculate f Y = R f X,Y dx explicitly. 1.1 Example Let X, Y f X,Y = e - x/y e - y y · 1 { x,y (0 , ) } . Find the conditional expectation E ( X | Y ). Well: 1
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f Y = Z 0 e - x/y e - y y dx = e - y Z 0 1 y e - x/y dx = e - y Which yields... f X | Y = e - x/y e - y y · 1 e - y = 1 y e - x/y So, the conditional expectation is: E ( X | Y = y ) = Z 0 x y e - x/y dx But this is just the expectation of an exponential random variable with parameter λ = 1 y . So E ( X | Y ) = y . 2 Computing Expectations by Conditioning Recall: E ( X ) = E ( E ( X | Y )); the expectation of the conditional expectation is the original, or unconditioned expectation. We saw a proof of this in an earlier lecture. This fact allows us to compute expectations by conditioning on any random variable (which makes the computation easy). 2.1 Example: Expectation of a Random Sum Let X 1 , ..., X N with N a random number and X i N . Find the expectation of the random sum E ( N i =1 X i ). Condition on N : 2
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E ( N X i =1 X i ) = E [ E ( N X i =1 X i | N = n )] = E [ N · E ( X )] = E ( N ) · E ( X ) 2.2 Example: ρ in N 2 is correlation Let X, Y N 2 ( μ X , μ Y , σ 2 X , σ 2 Y , ρ ), the bivariate normal distribution. We’ve defined the correlation ρ X,Y = E ( XY ) - μ X μ Y σ X σ Y
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