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lect20

# lect20 - CONDITIONAL EXPECTATION CONDITIONAL VARIANCE...

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CONDITIONAL EXPECTATION CONDITIONAL VARIANCE CONDITIONAL EXPECTATION AS BEST PREDICTION 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 CONDITIONAL EXPECTATION AS BEST PREDICTION 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
CONDITIONAL EXPECTATION CONDITIONAL VARIANCE CONDITIONAL EXPECTATION AS BEST PREDICTION 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 , μ y , σ x , σ 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 CONDITIONAL EXPECTATION AS BEST PREDICTION 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 ) .
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