Summary of Properties of Expectation, Variance, and Covariance

Summary of Properties of Expectation, Variance, and Covariance

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STAT 230 Summary of Expectation, Variance, and Covariance Rules Variance and Covariance deFnitions: 1. Var( X )= E ( ( X - E ( X )) 2 ) = E ( X 2 ) - E ( X ) 2 . 2. Cov( X,Y )= E (( X - E ( X ))( Y - E ( Y ))) = E ( XY ) - E ( X ) E ( Y ). Expectation Rules: 1. E ( a )= a for any constant a . 2. E ( aX )= aE ( X ) for any constant a . 3. E ( X + Y )= E ( X )+ E ( Y ) for any two r.v.’s. generalization: E ( n j =1 ( a j X j + b j )) = n j =1 a j E ( X j )+ n j =1 b j . Variance Rules: 1. Var( a ) = 0 and Var( X + a ) = Var( X ) for any constant a . 2. Var( aX )= a 2 Var( X ) for any constant a . 3. Var( X + Y ) = Var( X ) + Var( Y )+2Cov ( X,Y ) for any two r.v.’s. generalizations: 1. Var( aX + bY )= a 2 Var( X )+ b 2 Var( Y )+2 ab Cov( X,Y ). 2. Var( n j =1 ( a j X j + b j )) = n j =1 a 2 j Var( X j )+2 n - 1 i =1
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Unformatted text preview: a i a j Cov( X i ,X j ). Covariance Rules: 1. Cov( X,Y ) = Cov( Y,X ) for any two r.v.s. 2. Cov( X,Y ) = 0 if X and Y are independent. 3. Cov( X,a ) = 0 for any constant a . 4. Cov( aX,bY ) = ab Cov( X,Y ) for any constants a and b . 5. Cov( X,Y + Z ) = Cov( X,Y ) + Cov( X,Z ). Similarly Cov( X + Y,Z ) = Cov( X,Z ) + Cov( Y,Z ). generalization: Cov( aX + bY,cW + dZ ) = ac Cov( X,W ) + ad Cov( X,Z ) + bc Cov( Y,W ) + bd Cov( Y,Z )....
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