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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.
 Spring '08
 Zahrn
 The Land

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