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Unformatted text preview: Stat/219 Math 136  Stochastic Processes Notes on Section 4.1.2 1 Independence, Uncorrelatedness, and Something in Between Suppose ( X,Y ) is a random vector defined on some probability space (Ω , F , IP) with IE( X 2 ) < ∞ , IE( Y 2 ) < ∞ . The following implications are true. IE( X  Y ) = IE( X ) ր ց X and Y are independent X and Y are uncorrelated (1 A ) ց ր IE( Y  X ) = IE( Y ) For a proof of the first statements on the left above, assume X and Y are independent. Then the constant IE( X ) is measurable with respect to σ ( Y ). Since X and σ ( Y ) are independent, for any G ∈ σ ( Y ) we have integraldisplay G IE( X ) d IP = IE( X )IE( I G ) = IE( XI G ) by independence = integraldisplay G Xd IP Thus IE( X  Y ) = IE( X ). Similarly we have IE( Y  X ) = IE( Y ). Next we prove the statements on the right side of (1A). Suppose that IE( Y  X ) = IE( Y ). Using the tower property and then taking out what is known, we have IE( XY ) = IE(IE( XY  X )) = IE( X (IE( Y  X )) = IE( X )IE( Y ). Thus X and Y are uncorrelated. Similarly, IE( X  Y ) = IE( X ) implies that X and Y are uncorrelated. However, the converse implications are not true in general. Counterexample 1: X and Y uncorrelated does not imply IE( X  Y ) = IE( X ) Let Ω = {− 1 , , 1 } with P ( { ω } ) = 1 / 3 for each ω ∈ Ω. Let Y ( ω ) = ω and X ( ω ) = I { } ( ω ). Then XY = 0 so IE( XY ) = 0. Also, IE( Y ) = 0, and so IE( XY ) = 0 = IE( X )IE( Y ); that is, X and Y are uncorrelated. However, since X is measurable with respect to σ ( Y ) we have IE( X  Y ) = X which is never equal to IE( X ) = 1 / 3....
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This note was uploaded on 11/08/2009 for the course STAT 219 taught by Professor 2 during the Fall '08 term at Stanford.
 Fall '08
 2
 Probability

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