Download Moment generating functions

The strong law of large numbers
• Theorem. Let X1, X2, ... be a sequence of independent and
identically distributed random variables, each having the
same finite mean µ. Then, with probability 1,
X1  X 2  ...  X n
 μ as n  .
n
Proof. Left for graduate school.
• Example. Suppose a sequence of independent trials of an
experiment is performed. Let E be a fixed event of the
experiment and let P(E) be the probability that E occurs on
any particular trial. Let Xi = 1 if E occurs on the ith trial
and let Xi = 0 otherwise. Note that E[Xi] = P(E). By the
above theorem, except for a set of probability zero,
X1  X 2  ...  X n
 P(E) as n  .
n
The central limit theorem
• Theorem. Let X1, X2, ... be a sequence of independent and
identically distributed random variables, each having the
same finite mean µ and the same finite variance 2. Then
the distribution of
X1  X 2  ...  X n  nμ
 n
tends to the standard normal as n . That is , for all real
numbers a,
P
X1  X 2  ...  X n  nμ
 n
 a  (a) as n  ,
where  is the c.d.f. for the standard normal distribution.
An application of the central limit theorem
• Problem. Let Xi, i = 1, 2, ... , 10 be independent r.v.’s, each
uniformly distributed over (0, 1). Calculate an
approximation to
 10
P X i  6 .
 i 1

• Solution. Since E[Xi] = 1/2 and Var(Xi) = 1/12, we have
by the central limit theorem,
 10
 Xi  5
10
65
 i 1
P
X i  6  P

10( 112)
i 1
 10( 112)



 1   ( 1.2 )  0.1367