Copyright
c
2009 by Karl Sigman
Conditional expectation
Here we review some basic properties of conditional expectation that are useful for doing
computations and give several examples to help the reader memorize these properties.
(A more rigorous account can be found, for example, in Karlin and Taylor, Pages 59 in
Ch. 1 and then Pages 302–305 in Ch. 6.)
Recall for two rvs
X
and
Y
that
E
(
X

Y
) is itself a rv and is a function of
Y
, say
g
(
Y
); so then for example
E
(
X

Y
=
i
) =
g
(
i
). The idea is that besides the part of
X
determined by
Y
, the rest of
X
is averaged out with the expected value. By treating
Y
as
if it was a constant, and then integrating (averaging) out the rest yields the conditional
expectation. If
X
is determined by
Y
(for example
X
=
Y
or some function of
Y
), then
E
(
X

Y
) =
X
; nothing has been averaged out.
Basic properties
1.
E
(
X
) =
E
[
E
(
X

Y
)] for any rv
Y
. For example suppose
X
=
∑
N
n
=1
U
n
, where the
U
i
are iid and independent of the rv
N
. Letting
Y
=
N
yields
E
(
X

Y
) =
NE
(
U
),
and thus
E
[
E
(
X

Y
)] =
E
(
N
)
E
(
U
). We then conclude that
E
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 Summer '10
 KarlSigma
 Probability theory, Conditional expectation, Y, Karl Sigman

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