9.1 Example_Class_7_solution

9.1 Example_Class_7_solution - THE UNIVERSITY OF HONG KONG...

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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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(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 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 X i =1 a i X i , where a i ’s are constants. Then, E ( Y ) = n X i =1 aE ( X i ) V ar ( Y ) = n X i =1 V ar ( a i X i ) = n X i =1 a 2 i V ar ( X i ) M Y ( t ) = n Y i =1 M X i ( a i t ) Problems Problem 1. Let X and Y be two discrete random variables with joint probability mass function shown in the table X p X,Y ( x, y ) -1 0 1 -1 0.15 0.05 0.1 Y 0 0.1 0.1 0.2 1 0.15 0.05 0.1 (a)Calculate the covariance and correlation coefficient between X and Y. (b)Determine whether X and Y are independent.
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.

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9.1 Example_Class_7_solution - THE UNIVERSITY OF HONG KONG...

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