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Expected Value and Moments
Charles B. Moss
August 28, 2010
I. Expected Value
A. The random variable can also be described using a statistic.
1. One basic statistic encountered by students in statistics courses
is the mean of a random variable.
2.
Defnition 2.12
p28
The expected value (expectation or
mean) of a discrete random variable
X
, denoted
E
[
X
], is
deFned as
E
[
X
]=
X
x
i
∈
X
x
i
P
[
x
i
](
1
)
3.
Defnition 2.13
p28
The expected value of a continuous
random variable X is then deFned as
E
[
X
Z
x
∈
X
xf
(
x
)
dx
(2)
4. Taking the die roll as an example, if we let each side be equally
likely the expected value of the roll of a fair die is
E
[
x
6
X
i
=1
iP
[
i
]=1
1
6
+2
1
6
+3
1
6
+4
1
6
+5
1
6
+6
1
6
=3
1
2
(3)
6. This result points out an interesting fact about the expected
value, namely that the expected value need not be an element
of the sample set.
1
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View Full DocumentAEB 6182 Agricultural Risk Analysis and Decision Making
Professor Charles B. Moss
Lecture V
Fall 2010
7. Suppose we weight the die so that it is no longer fair. Specif
ically, assume that
P
[
i
]=1
/
9for
i
=
{
1
,
2
,
5
,
6
}
,
P
[3] = 3
/
9,
and
P
[4] = 2
/
9 , then the expected value becomes
E
[
X
]=
6
X
i
=1
iP
[
i
1
9
+2
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