Unformatted text preview: particular when B = ⌦ we have EY = Pk j =1 P (Aj )
P (B )
E (Y Aj ) · P (Aj ). 161 5.2. EXAMPLES, BASIC PROPERTIES Proof. Using the deﬁnition of conditional expectation, Lemma 5.3, then doing
some arithmetic and using the deﬁnition again, we have
E (Y B ) = E (Y ; B )/P (B ) =
= k
X E (Y ; Aj )/P (B ) j =1 k
k
X E (Y ; Aj ) P (Aj ) X
P (Aj )
·
=
E (Y Aj ) ·
P (Aj )
P (B )
P (B )
j =1
j =1 which proves the desired result.
In the discussion in this section we have concentrated on the properties of
conditional expectation given a single set A. To connect with more advanced
treatments, we note that given a partition A = {A1 , . . . An } of the sample space,
(i.e., disjoint sets whose union in ⌦) then the conditional expectation given the
partition is a random variable:
E (X A) = E (X Ai ) on Ai In this setting, Lemma 5.4 says
E [E (X A)] = EX
i.e., the random variable E (X A) has the same expected value as X . Lemma
5.1 says that if X is constant on each part of the partition then
E...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.
 Spring '10
 DURRETT
 The Land

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