Page 217 - Fundamentals of Probability and Statistics for Engineers

200 Fundamentals of Probability and Statistics for Engineers

Expanding X "t) in a MacLaurin series as indicated by Equation (4.49), we
can write

" #) n
2 2 2
jmt mjt ‡ m † jt
Z t†ˆ exp 1 ‡ ‡ ‡
n 1=2 n 1=2 2 n 1=2
n
2 2
t t
ˆ 1 ‡ o 7:18†
2n n
t
2
! exp ; as n !1:
2
In the last step we have used the elementary identity
c n

lim 1 ‡ ˆ e c 7:19†
n!1 n
for any real c.
Comparing the result given by Equation (7.18) with the form of the char-
acteristic function of a normal random variable given by Equation (7.12), we
see that Z (t) approaches the characteristic function of the zero-mean, unit-
variance normal distribution. The proof is thus complete.
As we mentioned earlier, this result is a somewhat restrictive version of the
central limit theorem. It can be extended in several directions, including cases
in which Y is a sum of dependent as well as nonidentically distributed random
variables.
The central limit theorem describes a very general class of random phenom-
ena for which distributions can be approximated by the normal distribution. In
words, when the randomness in a physical phenomenon is the cumulation of
many small additive random effects, it tends to a normal distribution irres-
pective of the distributions of individual effects. For example, the gasoline
consumption of all automobiles of a particular brand, supposedly manufac-
tured under identical processes, differs from one automobile to another. This
randomness stems from a wide variety of sources, including, among other
things: inherent inaccuracies in manufacturing processes, nonuniformities in
materials used, differences in weight and other specifications, difference in
gasoline quality, and different driver behavior. If one accepts the fact that each
of these differences contribute to the randomness in gasoline consumption,
the central limit theorem tells us that it tends to a normal distribution. By
the same reasoning, temperature variations in a room, readout errors asso-
ciated with an instrument, target errors of a certain weapon, and so on can also
be reasonably approximated by normal distributions.
Let us also mention that, in view of the central limit theorem, our result in
Example 4.17 (page 106) concerning a one-dimensional random walk should be

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