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[
]
[
]
[
]
[
]
+=
222
2)1( xExxExESEn
ii
[
]
[
]
[
]
[
]
+= 12)1(
222
xExxExESEn
ii
Now, if you think about it, it's clear that
= xnx
i
, so we can rewrite the middle term on
the RHS in terms of :
[
]
[
]
[]
[
]
+= 1)(2)1(
222
xExnxExESEn
i
[
]
[
]
[
]
222
)1( xnExESEn
i
=
[
]
[
]
[
]
222
)1( xnExnESEn
i
=
[] [] []
222
1
xExESE
n
n
i
=
Let's write that again as a numbered equation:
[
]
[
]
[
]
222
1
xExESE
n
n
i
=
(1)
Unfortunately, the expected value of the square of something is not equal to the square of
the expected value, so we seem to have hit an impasse with both terms on the RHS. But,
we're not out of tricks yet. Each of those terms is an expected value of something
squared: a second moment. Let's use the trick about moments that we saw above.
First, let Y be the random variable defined by the sample mean,
. We're trying to figure
out the expected value of its square.
[] []
[] []
2
22
var YEYxEYE +==
[] []
222
1
var
μ
+
==
i
x
n
xEYE
[] []
[]
2
2
22
var
1
μ
+==
i
x
n
xEYE
Sample Variance 2