Page 216 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 199

Let us note in passing that
2 , the coefficient of excess, defined by Equation
(4.12), for a normal distribution is zero. Hence, 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 fX n g be a sequence of mutually
independent and identically distributed random variables with means m and
variances 2 .Let

n
X
Y ˆ X j ; 7:14†
jˆ1

and let the normalized random variable Z be defined as

Y nm†
Z ˆ : 7:15†
n 1=2

Then the probability distribution function of Z , F Z (z), converges to N (0, 1) as
n !1 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 n 1/2 . Hence,
Z is simply the standardized random variable Y with zero mean and unit
standard deviation. In terms of characteristic functions X (t) of random vari-
ables X j , the characteristic function of Y is simply

n
Y t†ˆ ‰ X t†Š : 7:16†

Consequently, Z possesses the characteristic function

n
jmt t
Z t†ˆ exp X : 7:17†
n 1=2 n 1=2

TLFeBOOK