Page 116 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 99
4.5.1 GENERATION OF MOMENTS

One of the important uses of characteristic functions is in the determination of
the moments of a random variable. Expanding X (t) as a MacLaurin series, we
see that (suppressing the subscript X for convenience)

t 2 n† t n
00
0
t†ˆ 0†‡ 0†t ‡ 0† ‡ ‡ 0† ‡ ... ; 4:49†
2 n!
where the primes denote derivatives. The coefficients of this power series are,
from Equation (4.47),

1
Z 9
0†ˆ f x†dx ˆ 1; >
X >
>
1 >
>
>
>
>
Z 1 >
>
0
0†ˆ d t† ˆ jxf x†dx ˆ j 1 ; >
>
X
>
dt >
=
tˆ0 1
4:50†
. >
. >
>
. >
>
>
>
>
>
n Z 1 >
>
n
>
n†
n n
0†ˆ d t† ˆ j x f x†dx ˆ j n : ;
>
>
X
dt n
tˆ0 1
Thus,
1 n
X jt† n
t†ˆ 1 ‡ : 4:51†
n!
nˆ1
The same results are obtained when X is discrete.
Equation (4.51) shows that moments of all orders, if they exist, are contained

in the expansion of (t), and these moments can be found from (t) through

differentiation. Specifically, Equations (4.50) give
n n†
n ˆ j 0†; n ˆ 1; 2; ... : 4:52†
Example 4.14. Problem: determine (t), the mean, and the variance of a

random variable X if it has the binomial distribution
n k n k

p k†ˆ p 1 p† ; k ˆ 0; 1; ... ; n:
X
k
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