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Expected Value and Variance
*Classic Probability asserts that the
expected value
of a random variable is the longrun average value
of the random variable in independent observations.
Expected value
The expected value of a discrete random variable X, denoted E[X], is defined by
E[X]=
x
X
x
p
x
)
(
*
In words, the expected value of a discrete random variable is a weighted average of its possible values,
and the weight used is its probability.
Sometimes, the expected value is referred to as the expectation, the mean, or the first moment.
Also, sometimes it is denoted as
X
.
For a function of x g(x), we have the following:
E[g(X)]=
( )*
( )
X
x
g x
p
x
.
So, if we define g(x) =
2
x
, we have E[
2
x
] =
2
*
( )
X
x
x
p
x
.
Siblings (x)
Probability
)
(
x
p
X
X*
)
(
x
p
X
0
.200
1
.425
2
.275
3
.075
4
.025
1.000
As we can see, the expected value of our sibling example is __________. This implies that if we
randomly select a student then the expected number of siblings they have is ___________.
Clearly a student can’t have
___________ siblings. However, if we continue to select randomly students,
the average number of siblings we obtain will be close to ___________.
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View Full Document Proposition 7.1
Consider a finite population and a variable defined on it. Suppose that a member is selected at random
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This note was uploaded on 02/26/2012 for the course STAT 225 taught by Professor Martin during the Fall '08 term at Purdue UniversityWest Lafayette.
 Fall '08
 MARTIN
 Probability, Variance

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