Example_class_9_slides0

# Example_class_9_slides0 - c bY d = sign ab Corr X Y The...

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Example class 9 STAT1301 Probability and Statistics I November 15, 2011

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Review:Covariance 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 i = 1 a i X i + c i , N j = 1 b j Y j + d j ) = m i = 1 n j = 1 a i b j Cov ( X i , Y j ) (b) Cov ( X , X ) = Var ( X ) (c) Var ( X + Y ) = Var ( X )+ Var ( Y )+ 2 Cov ( X , Y ) In general, Var ( n i = 1 X i ) = n i = 1 Var ( 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!)
Review: Correlation Coeﬃcient 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 Var ( X ) p Var ( Y ) Properties: (a) - 1 ρ 1 (b) ρ is invariant under linear transformation X and Y. That is, Corr ( aX +

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Unformatted text preview: 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. 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. Problem 1 Solution Problem 2 I Problem 2 Solution Problem 2 Solution Problem 3 Problem 3 Solution Problem 3 Solution Problem 3 Solution Problem 4 Problem 4 Solution Problem 4 Solution I Problem 4 Solution Problem 5 Problem 5 Solution Problem 5 Solution Problem 6 Problem 6 Solution Problem 6 Solution...
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Example_class_9_slides0 - c bY d = sign ab Corr X Y The...

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