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

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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 ) .
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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 ) .
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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 ) .
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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 ) .
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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 ) .
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Conclusion : The relation Var( X + Y ) = Var X + Var Y holds if and only if Cov( X, Y ) = 0. In particular, if X and Y are independent, then Var( X + Y ) = Var X + Var Y.
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Application: diversification and risk I am considering investing in stocks, and I am looking at two stocks,
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