ISyE 2027 Probability with Applications
Polly B. He
Fall10, Week 15
Reading: Section 13.13, 14.12
The Law of Large Numbers
Intuitively, we view the probability of a certain outcome as the proportion of times that this
outcome occurs when the experiment is repeated many times over a long period of time.
Mathematically, we deﬁne probability as the value, say, of a p.m.f. for a random variable
representing the outcome. How can we be sure that these two are consistent?
Averages of a Sequence of Random Variables
Suppose
X
1
,X
2
,...,X
n
are
n
independent, identically distributed random variables (iden
tically distributed means these random variables have the same distribution). Then the
average of this sequence is
¯
X
n
=
X
1
+
X
2
+
···
+
X
n
n
If
¯
X
n
is the average of
n
independent random variables with the same expectation
µ
and
variance
σ
2
, then
E[
¯
X
n
] =
µ
and
Var(
¯
X
n
) =
σ
2
n
Chebyshev’s Inequality
The following inequality provides a bound for the probability that the random variable
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 Fall '08
 Zahrn
 Central Limit Theorem, Probability theory, Yi, p.m.f., Polly B. He Fall10

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