NotesSep24 - Linearity: If X and Y are random variables and...

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Unformatted text preview: Linearity: If X and Y are random variables and a is a number, Cov( aX,Y ) = a Cov( X,Y ) . Symmetry: Cov( X,Y ) = Cov( Y,X ) . Covariance and variance: For any random variable X Cov( X,X ) = Var( X ) . Additivity rule: Let X 1 ,X 2 and Y be ran- dom variables. Then Cov( X 1 + X 2 ,Y ) = Cov( X 1 ,Y ) + Cov( X 2 ,Y ) . The general additivity rule ; If X 1 ,X 2 ,...,X n and Y are random variables. Then Cov n X i =1 X i ,Y = n X i =1 Cov( X i ,Y ) . Additivity in both arguments : For any random variables X 1 ,...,X n and Y 1 ,...,Y k Cov n X i =1 X i , k X j =1 Y j = n X i =1 k X j =1 Cov( X i ,Y j ) . The correct formula for the variance of a sum . If X and Y are random variables, then it is NOT necessarily true that Var( X + Y ) = Var X + Var Y. The correct formula : Var( X + Y ) = Cov( X + Y,X + Y ) = Var X + Var Y + 2Cov( X,Y ) . Conclusion : The relation Var( X + Y ) = Var X + Var Y holds if and only if Cov( X,Y ) = 0....
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NotesSep24 - Linearity: If X and Y are random variables and...

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