prob2_3 - (ii) The correlation coefficient is a number...

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Unformatted text preview: (ii) The correlation coefficient is a number between- 1 and +1 this follows again from the Cauchy-Schwarz inequality. (iii) If Y = aX + b holds with a = 0 , then Var( Y ) = a 2 Var( X ) , E ( XY ) = aE ( X 2 ) + bE ( X ) and E ( X ) E ( Y ) = a ( E ( X ) ) 2 + bE ( X ) , thus Cov( X, Y ) = a Var( X ) . It follows that ( X, Y ) = 1 for a > and ( X, Y ) =- 1 for a < . Linear dependence between X and Y hence implies that ( X, Y ) attains one of the extreme values. Theorem 4.17. Let ( X, Y ) be a two-dimensional random variable with Var( X ) > and Var( Y ) > . (i) If X and Y are independent, then E ( X Y ) = E ( X ) E ( Y ) and ( X, Y ) = 0 . (ii) If ( X, Y ) = 1 or, if ( X, Y ) =- 1 , then there exist real numbers a and b such that P ( Y = aX + b ) = 1 . The coefficient a has the same sign as ( X, Y ) . (iii) The mean squared error E ( ( Y- aX- b ) 2 ) of a random variable Y from a linear function aX + b of the random variable X is minimal if a = Cov( X, Y ) Var( X ) and b = E ( Y )- aE ( X ) . In this case it is given by E ( ( Y- aX- b ) 2 ) = ( 1- ( X, Y ) ) Var( Y ) . Proof. The proof of (i) will only be given under the addition assumption that either both random variables are discrete or both are continuous. If X and Y are discrete random variables, then E ( XY ) = ij x i y j p ij = ij x i y j p i p j = i x i p i j y j p j = E ( X ) E ( Y ) . If X and Y are continuously distributed with densities f 1 and f 2 , then by f ( x, y ) = f 1 ( x ) f 2 ( y ) , ( x, y ) R 2 , a density f of ( X, Y ) is given. Then: E ( XY ) = - - x y f 1 ( x ) f 2 ( y ) d y d x = - x f 1 ( x )d x - y f 2 ( y )d y = E ( X ) E ( Y ) . Let us now show (iii). We find E ( ( Y- aX- b ) 2 ) = Var( Y- aX- b ) + ( E ( Y- aX- b ) ) 2 = Var( Y ) + a 2 Var( X )- 2 a Cov( X, Y ) + ( E ( Y )- aE ( X )- b ) 2 ....
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This note was uploaded on 05/26/2010 for the course STAT 443 taught by Professor Yuliagel during the Winter '09 term at Waterloo.

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prob2_3 - (ii) The correlation coefficient is a number...

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