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100BHWS4

# 100BHWS4 - STAT 100B HW4 Solution Problem 1 Let b = Cov(X...

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STAT 100B HW4 Solution Problem 1 Let b = Cov ( X,Y ) /V ar ( X ) V ar ( Y - bX ) = V ar ( Y ) + b 2 V ar ( X ) - 2 bCov ( X,Y ) = V ar ( Y ) + Cov 2 ( X,Y ) V ar 2 ( X ) V ar ( X ) - 2 Cov ( X,Y ) V ar ( X ) Cov ( X ) = V ar ( Y ) - Cov 2 ( X,Y ) V ar ( X ) 0 Cov 2 ( X,Y ) V ar ( X ) V ar ( Y ) = corr ( X,Y ) 2 1 Problem 2 Let S ( λ ) = 1 /n h ( x i ) = ¯ h Denote λ T = λ ture By Taylor expansion S ( λ ) = ¯ h S ( λ T ) + ( λ - λ T ) S 0 ( λ T ) ˆ λ = λ T + ¯ h - S ( λ T ) S 0 ( λ T ) E ( ˆ λ ) = λ T + E ( ¯ h ) - s ( λ T ) S 0 ( λ T ) = λ T , ( E ( ¯ h ) = E (1 /n h ( x i )) = 1 n nE ( h ( x )) = S ( λ T )) V ar ( ˆ λ ) = V ar ( λ T + ¯ h - s ( λ T ) S 0 ( λ T ) ) = V ar ( ¯ h ) S 0 ( λ T ) 2 = 1 /n 2 V ar ( ( h ( x i ))) S 0 ( λ T ) 2 = 1 /n 2 nV ar ( h ( x )) S 0 ( λ T ) 2 S 0 ( λ ) = ∂λ R h ( x ) λe - λx dx = R h ( x ) ∂λ ( λe - λx ) dx = R h ( x )[ e - λx + λe - λx ( - x )] dx = R h ( x ) λe - λx (1 - x ) dx = E [ h ( x

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100BHWS4 - STAT 100B HW4 Solution Problem 1 Let b = Cov(X...

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