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A07ans - EXERCISES IN STATISTICS Series A No 7 1 Let x and...

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EXERCISES IN STATISTICS Series A, No. 7 1. Let x and y be jointly distributed random variables with conditional ex- pectations which can be written as E ( y | x ) = α + βx and E ( x | y ) = γ + δy . Express β and δ in terms of the moments of the joint distributions and show that β 1 . Answer. We have E ( y | x ) = α + βx with β = C ( x, y ) V ( x ) and α = E ( y ) - βE ( x ) . Likewise E ( x | y ) = γ + δy with δ = C ( x, y ) V ( y ) and γ = E ( x ) - βE ( y ) . On forming the product of the two slope coefficients we can invoke the Cauchy– Schwarz Inequality: βδ = C ( x, y ) 2 V ( x ) V ( y ) = ' Corr ( x, y ) 2 1 . Hence β 1 . 2. A marksman’s scores are a sequence of random variables y i with E ( y i ) = 90 and V ( y i ) = 16 for all i . The correlation between successive scores is 0.9, and the expectation of a score conditional upon the previous score is given by E ( y i | y i - 1 ) = α + βy i - 1 where α = (1 - β ) E ( y i ). Find the expected score given that the previous score was 80. Answer. Substituting for α = (1 - β ) E ( y i ). gives E ( y i | y i - 1 ) = α + βy i - 1 = (1 - β ) E ( y i ) + βy i - 1 = E ( y i ) + β ' y i - 1 - E ( y i ) Also, since V ( y i ) = V ( y i - 1 ), we have Corr ( y, y i - 1 ) = C ( y i , y i - 1 ) p V ( y i ) V ( y i - 1 ) = C ( y i , y i - 1 ) V ( y i - 1 ) = β. 1
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SERIES A No.7, ANSWERS With E ( y i ) = E ( y i - 1 ) = 90, Corr ( y i , y i - 1 ) = 9 / 10, and y i - 1 = 80, this gives E ( y i | y i - 1 ) = 90 + 9 10 (80 - 90) = 81 .
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