1
LAW OF LARGE NUMBERS
Lecture 14
ORIE3500/5500 Summer2009 Chen
Class Today
•
Law of Large Numbers
•
Convergence
•
Normal (Gaussian) Distribution
1
Law of Large Numbers
The law of large numbers or l.l.n. is one of the most important theorems in
probability and is the backbone of most statistical procedures.
Theorem.
If
X
1
, . . . , X
n
are independent and identically distributed(iid) with
mean
μ
, then the sample mean
¯
X
n
converges to the true mean
μ
as
n
in
creases, that is,
¯
X
n
→
μ, n
→ ∞
.
Before we try to see why we should expect this let us recall a few properties
of the the sample mean,
¯
X
n
=
X
1
+
· · ·
+
X
n
n
.
1. Expected value of the sample mean,
E
(
¯
X
n
)
=
E
‡
X
1
+
· · ·
+
X
n
n
·
=
1
n
E
(
X
1
+
· · ·
+
X
n
)
=
1
n
nE
(
X
1
) =
μ.
1
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NOTIONS OF CONVERGENCE
2. If
var
(
X
1
) =
σ
2
, then variance of the sample mean,
var
(
¯
X
n
)
=
var
‡
X
1
+
· · ·
+
X
n
n
·
=
1
n
2
var
(
X
1
+
· · ·
+
X
n
)
=
1
n
2
(
var
(
X
1
) +
· · ·
+
var
(
X
n
)) (by independence)
=
1
n
2
n
·
var
(
X
1
) =
σ
2
n
.
This means that the variance of the sample mean decreases as the sample
size increases. Recall that the variance of a random variable measures the
dispersion of the random variable about its mean.
So if the variance is
decreasing to 0, then the random variable is slowly shrinking to its mean.
It becomes more and more concentrated around the population mean. The
Chebyshev’s inequality completes the argument. For any
² >
0
P
[

¯
X
n

μ

> ²
]
≤
var
(
¯
X
n
)
²
2
=
σ
2
/n
²
2
→
0
, n
→ ∞
.
This shows that whatever small number positive
²
we choose, the probability
that the sample mean is more than
²
distance away from the true mean goes
to zero. So we proved the LLN in the case when the variance of
X
1
is finite.
It can be proved without this assumption as well. Note that the statement
of LLN does not assume anything about the variance of
X
1
.
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 Spring '08
 BLAND
 Normal Distribution, Probability theory

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