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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.
The expected value of a discrete random variable X, denoted E[X], is defined by
Expected value
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
µ
.
For a function of x g(x), we have the following:
E[g(X)]=
()* ()
X
x
gx p x
∑
.
So, if we define g(x) =
2
x
, we have E[
2
x
] =
2
* ()
X
x
x
px
∑
.
Siblings (x)
Probability
)
(
x
p
X
X*
)
(
x
p
X
0
.200
.000
1
.425
.425
2
.275
.550
3
.075
.225
4
.025
.100
1.000
1.300
As we can see, the expected value of our sibling example is 1.3. This implies that if we randomly select a
student then the expected number of siblings they have is 1.3.
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View Full DocumentClearly a student can’t have 1.3 siblings. However, if we continue to select students at random, the
average number of siblings we obtain will be close to 1.3.
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 Spring '08
 MARTIN
 Probability, Variance

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