文档介绍：Chapter 9
Central Limit Theorem
Central Limit Theorem for Bernoulli Trials
The second fundamental theorem of probability is the Central Limit Theorem. This
theorem says that if Sn is the sum of n mutually independent random variables, then
the distribution function of Sn is well-approximated by a certain type of continuous
function known as a normal density function, which is given by the formula
1
(x
)2/(2
2)
f
,
(x)=√ e ,
2
as we have seen in Chapter . In this section, we will deal only with the case that
= 0 and
= 1. We will call this particular normal density function the standard
normal density, and we will denote it by
(x):
1 2
(x)=√ e
x /2 .
2
A graph of this function is given in Figure . It can be shown that the area under
any normal density equals 1.
The Central Limit Theorem tells us, quite generally, what happens when we
have the sum of a large number of independent random variables each of which con-
tributes a small amount to the total. In this section we shall discuss this theorem
as it applies to the Bernoulli trials and in Section we shall consider more general
processes. We will discuss the theorem in the case that the individual random vari-
ables are identically distributed, but the theorem is true, under certain conditions,
even if the individual random variables have di
erent distributions.
Bernoulli Trials
Consider a Bernoulli trials process with probability p for ess on each trial.
Let Xi = 1 or 0 according as the ith e is a ess or failure, and let
Sn = X1 + X2 +
+ Xn. Then Sn is the number of esses in n trials. We know
that Sn has as its distribution the binomial probabilities b(n, p, j). In Section ,
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326 CHAPTER 9. CENTRAL LIMIT THEOREM
0
-4 -2 0 2 4
Figure : Standard normal density.
we plotted these distributions for p = .3 and p = .5 for various values of n (see
Figure ).
We note that the maximum values of the distributions appea