Fundamentals of Probability and Statistics for Engineers

Let us note in passing that 2 , the coefficient of excess, defined by Equation
(4.12), for a normal distribution is zero. H ence, it is used as the reference
distribution for 2.

7.2.1 The Central Limit Theorem

The great practical importance associated with the normal distribution
stems from the powerful central limit theorem stated below (Theorem 7.1).
Instead of giving the theorem in its entire generality, it serves our
purposes quite well by stating a more restricted version attributable to
Lindberg (1922).

Theorem 7.1: the central limit theorem.Let be a sequence of mutually
independent and id entically distributed random variables with means m and
variances^2 .Let

and let the normalized random variable Z be defined as

Then the probability distribution function of Z,FZ (z), converges to N(0, 1) as
for every fixed z.
Proof of Theorem 7.1:We first remark that, following our discussion in
Section 4.4 on moments of sums of random variables, random variable Y
defined by Equation (7.14) has mean nm and standard deviation n1/2. Hence,
Z is simply the standardized random variable Y with zero mean and unit
standard deviation. In terms of characteristic functions X(t)ofrandomvari-
ables Xj, the characteristic function of Y is simply

Consequently, Z possesses the characteristic function

Some Important Continuous Distributions 199

fXng



Yˆ

Xn

jˆ 1

Xj; … 7 : 14 †

n!1

Zˆ

…Ynm†

n^1 =^2 

: … 7 : 15 †



Y…t†ˆ‰X…t†Šn: … 7 : 16 †

Z…t†ˆexp 

jmt

n^1 =^2 



X

t

n^1 =^2 

n

: … 7 : 17 †