Page 215 - Fundamentals of Probability and Statistics for Engineers

198 Fundamentals of Probability and Statistics for Engineers

graph of f (x) in this particular case is the well-known bell-shaped curve,
X
symmetrical about the origin [Figure 7.6(a)].
Let us determine the mean and variance of X. By definition, the mean of X,
EfXg , is given by

" #
Z 1 1 Z 1 x m† 2
EfXgˆ xf x†dx ˆ 1=2 x exp 2 dx;
X
1 2 † 1 2
which yields

EfXgˆ m:

Similarly, we can show that

2
var X†ˆ : 7:11†
We thus see that the two parameters m and in the probability distribution
are, respectively, the mean and standard derivation of X. This observation
justifies our choice of these special symbols for them and it also points out
an important property of the normal distribution – that is, the knowledge of
its mean and variance completely characterizes a normal distribution. Since the
normal distribution will be referred to frequently in our discussion, it is some-
times represented by the simple notation N( , 2 ). Thus, for example,
m
X : N(0, 9) implies that X has the pdf given by Equation (7.9) with m ˆ 0 and
ˆ 3.
Higher-order moments of X also take simple forms and can be derived in
a straightforward fashion. Let us first state that, following the definition of
characteristic functions discussed in Section 4.5, the characteristic function of a
normal random variable X is

" 2 #
1
1 x m† Š
Z
jtX
X t†ˆ Efe gˆ 1=2 exp jtx 2 dx
2 † 1 2
2 2
t
ˆ exp jmt ; 7:12†
2
The moments of X of any order can now be found from the above through
differentiation. Expressed in terms of central moments, the use of Equation
(4.52) gives us

0; if n is odd;
n ˆ n 7:13†
1 3† n 1† ; if n is even.

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