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exists and ∂y F Y | X ( y | x ) = f XY ( x, y ) f X ( x ) . Also, E [ Y | X = x ] = Z -∞ yf Y | X ( y | x ) dy. Analogously for continuous random variables E [ Y | X = x k ] = X l y l p Y | X ( y l | x k ) . A more general definition of conditional expectation We will explore conditional expectation in terms of probability spaces. Suppos G is a sub σ -field of F . A function g : Ω R is “measurable with respect to G ” if { ω Ω : g ( ω ) B } ∈ G for all B ∈ B (Any such function g () would be a r.v. too. But this g () is more restricted.) We will now define “conditioning on a σ -field.” Suppose Y is a r.v. with E [ | Y | 2 ] < and G is a sub σ -field of F . Then E [ Y |G ] is any G -measurable random variable such that Z A E [ Y |G ] dP = Z A Y dP for all A ∈ G . If Y itself were G measurable it would be its own conditional expectation. Example 8 Let ( ω, F ) = ( R , B ) . Let G = { ( -∞ , 0) , [0 , ) , , R } . (This is a σ -field.) Let Y be a r.v. Then E [ Y |G ] is any G -measurable r.v. satisfying Z R E [ Y |G ] dP = E [ Y ] Z ( -∞ , 0) E [ Y |G ] dP = Z ( -∞ , 0) Y dP Z [0 , ) E [ Y |G ] dP = Z [0 , ) Y dP. Note: to be measurable on G means being constant on ( -∞ , 0) and [0 , ) . The G -measurable r.v.s in this case are simple functions: g ( ω ) = ( b + ω 0 b - ω < 0 so b - P (( -∞ , 0)) = Z ( -∞ , 0) Y dP b + P ([0 , )) = Z [0 , ) Y dP This gives us two equations in two unknowns. b + = R [0 , ) Y dP R [0 , ) dP b - = R -∞ , 0) Y dP R ( -∞ , )) dP
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ECE 6010: Lecture 2 – More on Random Variables 16 2 Definition 16 If X is an r.v., define σ ( X ) (the σ -field generated by X ) to be {{ ω Ω : X ( ω ) B } for B ∈ B} . 2 Fact: A r.v. Y is measurable with respect to σ ( X ) if and only if there is a measurable function g : R R such that Y = g ( X ) . We now define conditional expectation with respect to a σ -field: Definition 17 If X and Y are r.v.s with E [ | Y | ] < , we define E [ Y | X ] = E [ Y | σ ( X )] 2 Properties: 1. By the fact stated above, we can write E [ Y | X ] = g ( x ) for some function g , g ( x ) = E [ Y | X = x ] . 2. E [ Y ] = E [ E [ Y | X ]] 3. If Y itself is G -measurable, then E [ Y |G ] = Y . 4. E [ αY 1 + βY 2 |G ] = αE [ Y 1 |G ] + βE [ Y 2 |G ] . 5. If Y 0 , then E [ Y |G ] 0 . 6. If E [ | Y | ] < and G ⊂ E ⊂ F , then E [ E [ Y |E ]] = E [ Y |G ] . Idea: If you first condition on a field that is less “course” than G you get a r.v. Then condition on G . Definition 18 Two σ -fields G and H are independent if P ( GH ) = P ( G ) P ( H ) for all G ∈ G and H ∈ H . 2 Note: X and Y are independent r.v.s iff σ ( X ) and σ ( Y ) are independent σ -fields. 6. If σ ( Y ) is independent of G then E [ Y |G ] = E [ Y ] . 7. If Y is G -measurable then E [ Y |G ] = E [ Y ] . So, for example, if G = σ ( X ) , and Y = g ( X ) for some g : R R (that is, Y is G -measurable) then E [ Y | X ] = Y = g ( X ) . More informally, E [ g ( x ) | X ] = g ( x ) .
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  • Fall '08
  • Stites,M
  • Probability theory, lim

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