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1
Weak Law of Large Numbers
•
Consider I.I.D. random variables X
1
, X
2
, .
..
X
i
have distribution
F
with E[X
i
] =
m
and Var(X
i
) =
s
2
Let
For any
e
> 0:
•
Proof:
By Chebyshev’s inequality:
0
)
(
n
X
P
e
n
n
X
X
X
E
X
E
...
2
1
]
[
n
n
n
X
X
X
X
2
...
2
1
Var
)
(
Var
0
)
(
2
2
n
n
X
P
n
i
i
X
n
X
1
1
Strong Law of Large Numbers
•
Consider I.I.D. random variables X
1
, X
2
, .
..
X
i
have distribution
F
with E[X
i
] =
Let
Strong Law
Weak Law, but not vice versa
Strong Law implies that for any
> 0, there are only a
finite number of values of
n
such that condition of
Weak Law:
holds.
n
i
i
X
n
X
1
1
1
...
lim
2
1
n
X
X
X
P
n
n
X
Intuitions and Misconceptions of LLN
•
Say we have repeated trials of an experiment
Let event E = some outcome of experiment
Let X
i
= 1 if E occurs on trial
i
, 0 otherwise
Strong Law of Large Numbers (Strong LLN) yields:
Recall first week of class:
Strong LLN justifies “frequency” notion of probability
Misconception arising from LLN:
o
Gambler’s fallacy: “I’m due for a win”
o
Consider being “due for a win” with repeated coin flips.
..
)
(
]
[
...
2
1
E
P
X
E
n
X
X
X
n
n
E
n
E
P
n
)
(
lim
)
(
La Loi des Grands Nombres
•
History of the Law of Large Numbers
1713: Weak LLN described by Jacob Bernoulli
1835: Poisson calls it “La Loi des Grands Nombres”
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 Spring '09

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