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

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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 ) 1 Moment-generating function: – Definition: M ( t ) = M X ( t ) = E( e tX ) – Computing moments via mgf’s: The derivates of M ( t ), evaluated at t = 0, give the successive “moments” of a random variable X : M (0) = 1, M (0) = E( X ), M (0) = E( X 2 ), M (0) = E( X 3 ), etc.
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