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09Stoch2

# 09Stoch2 - 1.2 CONDITIONAL EXPECTATION 13 1.2 Conditional...

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Unformatted text preview: 1.2. CONDITIONAL EXPECTATION 13 1.2 Conditional Expectation Let Λ ∈ F with P (Λ) > 0, we define P ( E | Λ) = P (Λ ∩ E ) P (Λ) where P (Λ) > . It follow that for a random vector ( X,Y ), P ( Y ≤ y | X = x ) =        P ( Y ≤ y, X = x ) P ( X = x ) , if P ( X = x ) > , , otherwise . Moreover if ( X,Y ) has joint density f ( x,y ), the conditional density of Y given X = x is f ( y | x ) =        f ( x,y ) f X ( x ) , if f X ( x ) > , , otherwise . where f X ( x ) = R ∞-∞ f ( x,y ) dy is the marginal density. The conditional expec- tation of Y given X = x is E ( Y | X = x ) = Z ∞-∞ yf ( y | x ) dy. Note that g ( x ) := E ( Y | X = x ) is a function on x , and hence g ( X ( · )) := E ( Y | X ( · )) is a r.v. on Ω . (1.2.1) In the following we have a more general consideration for the conditional expectation (and also the conditional probability): E ( Y |G ) where G is a sub- σ-field of F . First let us look at a special case where G is generated by a measurable partition { Λ n } n of Ω (each member in G is a union of { Λ n } n ). Let Y be an 14 CHAPTER 1. BASIC PROBABILITY THEORY integrable r.v., then E ( Y | Λ n ) = Z Ω Y ( ω ) dP Λ n ( ω ) = 1 P (Λ n ) Z Λ n Y ( ω ) dP ( ω ) . (1.2.2) (Here P Λ n ( · ) = P ( · ∩ Λ n ) P (Λ n ) is a probability measure for P (Λ n ) > 0). Consider the random variable (as in (1.2.1)) Z ( · ) = E ( Y |G )( · ) := X n E ( Y | Λ n ) χ Λ n ( · ) ∈ G . It is easy to see that if ω ∈ Λ n , then Z ( ω ) = E ( Y | Λ n ), and moreover Z Ω E ( Y |G ) dP = X n Z Λ n E ( Y |G ) dP = X n E ( Y | Λ n ) P (Λ n ) = Z Ω Y dP . We can also replace Ω by Λ ∈ G and obtain Z Λ E ( Y |G ) dP = Z Λ Y dP ∀ Λ ∈ G ....
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09Stoch2 - 1.2 CONDITIONAL EXPECTATION 13 1.2 Conditional...

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