1
CONDITIONAL EXPECTATION
Lecture 11
ORIE3500/5500 Summer2011 Li
1
Conditional Expectation
We have defined expectation of a random variable only in two cases: discrete
and continuous.
Expectation can be defined for all random variables but
that is outside the scope of this class. We will define conditional expectation
in these two cases only.
1. If (
X, Y
) is a discrete bivariate random vector with joint pmf
p
X,Y
(
x
i
, y
j
),
then the conditional expectation of
X
given
Y
=
y
j
is defined to be
E
(
X

Y
=
y
j
) =
X
i
x
i
p
X

Y
(
x
i

y
j
)
.
2. If (
X, Y
) is a continuous bivariate random vector with joint pdf
f
X,Y
(
x, y
),
then the conditional expectation of
X
given
Y
=
y
is defined to be
E
(
X

Y
=
y
) =
Z
∞
∞
xf
X

Y
(
x

y
)
dx.
Let a function be
g
(
Y
) =
E
(
X

Y
), observe that
g
(
Y
) is also a random
variable. We could define its expectation
E
(
g
(
Y
)) =
E
[
E
(
X

Y
=
y
)]
∑
j
E
[
X

Y
=
y
j
]
p
Y
(
y
j
)
Y
is discrete
R
∞
∞
E
[
X

Y
=
y
]
f
Y
(
y
)
dy
Y
is continuous
Example
Suppose that the joint density of
X
and
Y
is given by
f
(
x, y
) =
e

x/y
e

y
y
,
0
< x, y <
∞
. Compute
E
(
X

Y
=
y
).
Solution: First compute the conditional density
f
(
x

y
) and then
E
(
X

Y
=
y
).
Note that from Lecture 9 we know that
f
(
x

y
) =
f
(
x,y
)
f
Y
(
y
)
=
y

1
e

x/y
.
Therefore,
E
(
X

Y
=
y
) =
R
∞
0
xe

x/y
/ydx
=
y
R
∞
0
te

t
dt
=
y
.
1
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CONDITIONAL EXPECTATION
Law of Iterated Expectation:
E
(
X
) =
E
[
E
(
X

Y
)]
Proof in the discrete case: Note that
E
(
E
(
X

Y
))
=
X
y
E
(
X

Y
=
y
)
P
(
Y
=
y
) =
X
y
X
x
xP
(
X
=
x

Y
=
y
)
P
(
Y
=
y
)
=
X
y
X
x
xP
(
X
=
x, Y
=
y
) =
X
x
X
y
xP
(
X
=
x, Y
=
y
)
=
X
x
x
X
y
P
(
X
=
x, Y
=
y
) =
X
x
xP
(
X
=
x
) =
EX.
Intuition: To calculate
EX
, we may take a weighted average of the con
ditional expected value of
X
given
Y
=
y
, each of the terms
E
(
X

Y
=
y
)
being weighted by the probability of the event on which it is conditioned.
Example
Here is a game: there are two different coins. One is fair and the other one
has 0
.
9 chance to get tails when it is tossed. If you have 3
/
4 probability to
get the fair one and you will win one dollar if you get a head and nothing
otherwise. What is the expectation of the dollar you could win? Define the
random variable
X
as the dollar you could win, and define a random variable
Y
, which equals to one if you pick the fair coin and zero otherwise. Then we
have
E
[
X

Y
= 1] = 1
·
1
2
+ 0
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 Spring '11
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 Variance, Probability theory, 3 hours, 5 hours, Y

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