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Example_class_9_handout0 - aX c,bY d = sign ab Corr X,Y The...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 9 Review Covariance Let X and Y be random variables with mean μ x and 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 ( n X i =1 a i X i + c i , N X 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 i =1 X i ) = n i =1 V ar ( X i ) + 2 i<j Cov ( X i , Y 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 indepenent. (Very important!) Correlation Coefficient Let X and Y be two random variables. The correlation coefficient between X and Y, denoted as ρ xy is defined as Corr ( X, Y ) = Cov ( X, Y ) p V ar ( X ) p V ar ( Y ) Properties: (a) - 1 ρ 1 (b) ρ is invariant under linear transformation X and Y. That is, Corr
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Unformatted text preview: aX + c,bY + d ) = sign ( ab ) Corr ( X,Y ) The sign and the magnitude of ρ reveal the direction and strength of the linear relationship between X and Y. Sum of independent random variables Let X 1 ,X 2 ,...,X n be independent random variables, Y = ∑ n i =1 a i X i , where a i s are constants. Then, E ( Y ) = ∑ n i =1 aE ( X i ) V ar ( Y ) = ∑ n i =1 V ar ( a i X i ) = ∑ n i =1 a 2 i V ar ( X i ) M y ( t ) = Q n i =1 M x i ( a i t ) 1 Problems Problem 1 Let X and Y be two discrete random variables with joint probability mass function shown in the table. (a)Calculate the covariance and correlation coecient between X and Y. (b)Determine whether X and Y are independent. 2 3...
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