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Theorem 17.4
Suppose the
X
i
are i.i.d. Suppose
E
X
2
i
<
∞
. Let
μ
=
E
X
i
and
σ
2
= Var
X
i
. Then
P
±
a
≤
S
n

nμ
σ
√
n
≤
b
²
→
P
(
a
≤
Z
≤
b
)
for every
a
and
b
, where
Z
is a
N
(0
,
1)
.
The ratio on the left is (
S
n

E
S
n
)
/
√
Var
S
n
. We do not claim that this
ratio converges for any
ω
(in fact, it doesn’t), but that the probabilities
converge.
An example. If the
X
i
are i.i.d. Bernoulli random variables, so that
S
n
is
a binomial, this is just the normal approximation to the binomial.
An example. Suppose we roll a die 3600 times. Let
X
i
be the number
showing on the
i
th
roll. We know
S
n
/n
will be close to 3
.
5. What’s the
probability it diﬀers from 3
.
5 by more than 0
.
05?
Answer. We want
P
±³
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.
 Spring '10
 ansan

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