5 ln101 04 ln1 01 ln099 000397 2 2 vardk

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Unformatted text preview: λX )2 ] = E [Y 2 ] − 2λE [XY ] + λ2 E [X 2 ] E [XY ]2 = E [Y 2 ] − , E [X 2 ] (4.24) which implies that E [XY ]2 ≤ E [X 2 ]E [Y 2 ], which is equivalent to the Schwarz inequality. If P {Y = cX } = 1 for some c then equality holds in (4.23), and conversely, if equality holds in (4.23) then equality also holds in (4.24), so E [(Y − λX )2 ] = 0, and therefore P {Y = cX } = 1 for c = λ. Corollary 4.8.4 For two random variables X and Y, |Cov(X, Y )| ≤ Var(X )Var(Y ). Furthermore, if Var(X ) = 0 then equality holds if and only if Y = aX + b for some constants a and b. Consequently, if Var(X ) and Var(Y ) are not zero, so the correlation coefficient ρX,Y is well defined, then • |ρX,Y | ≤ 1, • ρX,Y = 1 if and only if Y = aX + b for some a, b with a > 0, and • ρX,Y = −1 if and only if Y = aX + b for some a, b with a < 0. Proof. The corollary follows by applying the Schwarz inequality to the random variables X − E [X ] and Y − E [Y ]. Example 4.8.5 Suppose n fair dice are independ...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.

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