lect20 - CONDITIONAL EXPECTATION CONDITIONAL VARIANCE...

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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE Outline CONDITIONAL EXPECTATION Computing Expectations by Conditioning Wald’s Identity CONDITIONAL VARIANCE Variance of a Sum of a Random Number of Random Variables CONDITIONAL EXPECTATION AS BEST PREDICTION 1 / 14 Xinghua Zheng Lect 20: Conditional Expectation/Variance; Best Prediction
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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE Computing Expectations by Conditioning Wald’s Identity CONDITIONAL EXPECTATION Let X and Y be two random variables. The conditional expectation E ( Y | X = x ) of Y given X = x is the expected value of the conditional distribution of Y given X = x : E ( Y | X = x ) = X y y · f Y | X ( y | x ) , in the discrete case, Z y y · f Y | X ( y | x ) dy , in the continuous case. (1) If X and Y are independent, then E ( Y | X = x ) = . 2 / 14 Xinghua Zheng Lect 20: Conditional Expectation/Variance; Best Prediction
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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE Computing Expectations by Conditioning Wald’s Identity EXAMPLES Suppose X Poisson ( λ 1 ) , Y Poisson ( λ 2 ) , and X and Y are independent. Put Z = X + Y . What’s the conditional expectation E ( X | Z = n ) ? The conditional distribution of X given Z = n is . Suppose X and Y are jointly normal with parameters μ x y , σ x y and ρ . What’s the conditional expectation E ( Y | X = x ) ? The conditional distribution of Y given X = x is . 3 / 14 Xinghua Zheng Lect 20: Conditional Expectation/Variance; Best Prediction
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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE Computing Expectations by Conditioning Wald’s Identity Some Useful Properties of Condition Expectation 1. Formula (1) generalizes to the case of computing the conditional expectation of a function of X and Y : suppose g ( x , y ) is a function, then the conditional expectation E ( g ( X , Y ) | X = x ) of g ( X , Y ) given X = x is E ( g ( X , Y ) | X = x ) = X y g ( x , y ) · f Y | X ( y | x ) , in the discrete case, Z y g ( x , y ) · f Y | X ( y | x ) dy , in the continuous case. 2. E ( g ( X ) | X = x ) = g ( x ) . 3. E ( g ( X ) h ( Y ) | X = x ) = g ( x ) · E ( h ( Y ) | X = x ) . 4 / 14 Xinghua Zheng Lect 20: Conditional Expectation/Variance; Best Prediction
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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE Computing Expectations by Conditioning Wald’s Identity COMPUTING EXPECTATIONS BY CONDITIONING For each x , let ϕ ( x ) := E ( Y | X
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lect20 - CONDITIONAL EXPECTATION CONDITIONAL VARIANCE...

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