variance - Math 461 Introduction to Probability A.J....

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Unformatted text preview: Math 461 Introduction to Probability A.J. Hildebrand Variance, covariance, correlation, moment-generating functions [In the Ross text, this is covered in Sections 7.4 and 7.7. See also the Chapter Summary on pp. 405–407.] • Variance: – Definition: Var( X ) = E( X 2 )- E ( X ) 2 (= E ( X- E ( X )) 2 ) – Properties: Var( c ) = 0, Var( cX ) = c 2 Var( X ), Var( X + c ) = Var( X ) • Covariance: – Definition: Cov( X,Y ) = E ( XY )- E ( X ) E ( Y )(= E ( X- E ( X ))( Y- E ( Y ))) – Properties: * Symmetry: Cov( X,Y ) = Cov( Y,X ) * Relation to variance: Var( X ) = Cov( X,X ), Var( X + Y ) = Var( X )+Var( Y )+2 Cov( X,Y ) * Bilinearity: Cov( cX,Y ) = Cov( X,cY ) = c Cov( X,Y ), Cov( X 1 + X 2 ,Y ) = Cov( X 1 ,Y ) + Cov( X 2 ,Y ), Cov( X,Y 1 + Y 2 ) = Cov( X,Y 1 ) + Cov( X,Y 2 ). * Product formula: Cov( ∑ n i =1 X i , ∑ m j =1 Y j ) = ∑ n i =1 ∑ m y =1 Cov( X i ,Y j ) • Correlation: – Definition: ρ ( X,Y ) = Cov( X,Y ) √ Var( X )Var( Y ) – Properties:- 1 ≤ ρ ( X,Y...
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This note was uploaded on 06/20/2011 for the course ECON 803 taught by Professor Pp during the Spring '11 term at Thammasat University.

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variance - Math 461 Introduction to Probability A.J....

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