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Unformatted text preview: ing Y by 1A sets the product = 0 on Ac and leaves the values
on A unchanged. Finally, we deﬁne the conditional expectation of Y given
A to be
E (Y A) = E (Y ; A)/P (A)
This is the expected value for the conditional probability deﬁned by
P (·A) = P (· \ A)/P (A)
Example 5.1. A simple but important special case arises when the random
variable Y and the set A are independent, i.e., for any set B we have
P (Y 2 B, A) = P (Y 2 B )P (A)
Noticing that this implies that P (Y 2 B, Ac ) = P (Y 2 B )P (Ac ) and comparing
with the deﬁnition of independence of random variables in (A.13), we see that
this holds if and only Y and 1A are independent, so Theorem A.1 implies
E (Y ; A) = E (Y 1A ) = EY · E 1A
159 160 CHAPTER 5. MARTINGALES and we have
E (Y A) = EY (5.1) It is easy to see from the deﬁnition that the integral over A is linear:
E (Y + Z ; A) = E (Y ; A) + E (Z ; A) (5.2) so dividing by P (A), conditional expectation also has this property
E (Y + Z A) = E (Y A) + E (Z A) (5.3) Here and in later formulas and theorems, we...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).
 Spring '10
 DURRETT
 The Land

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