1
EXPECTATION
Lecture 10
ORIE3500/5500 Summer2009 Chen
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Expectation
1
Expectation
1.1
Expectation of a Random Variable
Expectation
or
expected value
of a random variable denoted by
E
(
X
) of just
EX
is the weighted average of the values it takes, where the weight are the
chances of the random variable taking that value.
As the name suggests,
if we were to expect some value from a random variable to take, then this
would be the one. The expected value of a discrete random variable
X
, with
pmf
p
X
, is defined as
E
(
X
) =
X
i
x
i
p
X
(
x
i
)
.
For a continuous random variable
X
with density
f
X
, the expectation is
defined as
E
(
X
) =
Z
∞
∞
xf
X
(
x
)
dx.
Sometimes it is difficult to compute the expectation of a random vari
able by following the definition. There are alternate approaches to compute
expectation. The next example deals with one such case.
Example
If
X
is discrete and takes values 0
,
1
,
2
, . . .
, then we can reduce the expression
of expectation involving the cdf to
E
(
X
) =
∞
X
n
=0
P
[
X > n
] =
∞
X
n
=0
[1

F
X
(
n
)]
.
It is not difficult to see why this is true. As in the continuous case this
1
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1.1
Expectation of a Random Variable
1
EXPECTATION
involves changing of the order of summation.
E
(
X
)
=
∞
X
n
=0
np
X
(
n
) =
∞
X
n
=0
nP
[
X
=
n
]
=
P
[
X
= 1]
+
P
[
X
= 2] +
P
[
X
= 2]
+
P
[
X
= 3] +
P
[
X
= 3] +
P
[
X
= 3]
.
.
.
=
P
[
X
= 1] +
P
[
X
= 2] +
P
[
X
= 3] +
P
[
X
= 4] +
· · ·
+
P
[
X
= 2] +
P
[
X
= 3] +
P
[
X
= 4] +
· · ·
+
P
[
X
= 3] +
P
[
X
= 4] +
· · ·
.
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 Summer '08
 WEBER
 Probability theory, cdf FX

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