EECS 501
CENTRAL LIMIT THEOREM
Fall 2001
DEF:
rvs
{
x
1
, x
2
. . .
}
are
iidrv
with means
μ
and variances
σ
2
if:
1. The
x
i
are
independent
:
f
x
1
,x
2
...
(
X
1
, X
2
. . .
) =
f
x
1
(
X
1
)
f
x
2
(
X
2
)
. . .
2.
x
i
are
identically distributed
:
f
x
i
(
X
) =
f
x
(
X
)
, E
[
x
i
] =
μ, σ
2
x
i
=
σ
2
.
THM:
y
n
=
∑
n
i
=1
x
i
=sum of iidrvs
x
i
with
finite
means
μ
and variance
σ
2
.
Then:
E
[
y
n
] =
nμ
;
σ
2
y
n
=
nσ
2
; ˜
y
n
=
y
n

nμ
√
nσ
=
1
√
n
∑
n
i
=1
x
i

μ
σ
=
1
√
n
∑
n
i
=1
˜
x
i
.
Mean:
Let
m
n
=
y
n
n
=
1
n
∑
n
i
=1
x
i
=
mean
. Then
E
[
m
n
] =
μ
and
σ
2
m
n
=
σ
2
n
.
Proof:
All of these follow immediately from the basic properties of variance.
Note:
Variance of (sample) mean gets
smaller
with
n
! ”Regression to mean.”
While variance of
y
n
grows as
n
, variance of
y
n
n
”grows” as
1
n
→
0.
Note:
Does
not
mean that
m
n
”remembers” to correct deviations from
μ
!
DEF:
The
characteristic function
Φ
x
(
ω
) of rv
x
is
Φ
x
(
ω
) =
E
[
e
jωx
] =
R
∞
∞
f
x
(
X
)
e
jωX
dX
=
F{
f
x
(
X
)
}
(note sign).
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 Spring '10
 Lehman
 Central Limit Theorem, Normal Distribution, Variance, Probability theory, Cauchy distribution, φX

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