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9 Example Class 7

# 9 Example Class 7 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 7 Review Covariance Let X and Y be random variables with means μ x , μ y respectively. The covariance between X and Y, denoted by σ xy , is defined as Cov ( X, Y ) = E [( X - μ x )( Y - μ y )] = E ( XY ) - E ( X ) E ( Y ) Properties: (a) Cov ( aX + c, bY + d ) = abCov ( X, Y ) In general, Cov ( m i =1 a i X i + c i , n j =1 b j Y j + d j ) = m X i =1 n X j =1 a i b j Cov ( X i , Y j ) (b) Cov ( X, X ) = V ar ( X ) (c) V ar ( X + Y ) = V ar ( X ) + V ar ( Y ) + 2 Cov ( X, Y ) In general, V ar ( n X i =1 X i ) = n X i =1 V ar ( X i ) + 2 X i<j Cov ( X i , X j ) (d) If X and Y are independent, then Cov ( X, Y ) = 0. But Cov ( X, Y ) = 0 does not imply X and Y are independent. (Very important!!!) Correlation Coefficent Let X and Y be two random variables. The correlation coefficient between X and Y, denoted by ρ xy , is defined as Corr ( X, Y ) = Cov ( X, Y ) p V ar ( X ) p V ar ( Y ) 1

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Properties: (a) - 1 ρ 1. (b) ρ is invariant under linear transformation of X and Y. That is, Corr ( aX + c, bY + d ) = sign ( ab ) Corr ( X, Y ) The sign and the magnitude of ρ
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9 Example Class 7 - THE UNIVERSITY OF HONG KONG DEPARTMENT...

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