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Outline
Expected Value
Moments of a Distribution
Expected Value and Moments
Charles B. Moss
August 28, 2010
Charles B. Moss
Expected Value and Moments
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Expected Value
Moments of a Distribution
Expected Value
Empirical Moments
Applications
Bivariate Random Variables
Moments of a Distribution
Moments of the Uniform Distribution
Charles B. Moss
Expected Value and Moments
Outline
Expected Value
Moments of a Distribution
Empirical Moments
Applications
Bivariate Random Variables
Expected Value
I
The random variable can also be described using a statistic.
I
One basic statistic encountered by students in statistics
courses is the mean of a random variable.
I
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
)
I
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)
Charles B. Moss
Expected Value and Moments
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Expected Value
Moments of a Distribution
Empirical Moments
Applications
Bivariate Random Variables
I
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)
I
This result points out an interesting fact about the expected
value, namely that the expected value need not be an element
o
fthesamp
leset
.
Charles B. Moss
Expected Value and Moments
Outline
Expected Value
Moments of a Distribution
Empirical Moments
Applications
Bivariate Random Variables
I
Suppose we weight the die so that it is no longer fair.
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.
 Fall '08
 Weldon

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