Fundamentals of Probability and Statistics for Engineers

One of the important uses of ch aracteristic 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)

where the primes denote derivatives. The coefficients of this power series are,
from Equation (4.47),

Thus,

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

Ex ample 4. 14. Problem: determine (t), the mean, and the variance of a
random variable X if it has the binomial distribution

Expectations and Moments 99

… 0 †ˆ

Z 1

1

fX…x†dxˆ 1 ;

^0 … 0 †ˆ

d…t†

dt

tˆ 0

ˆ

Z 1

1

jxfX…x†dxˆj 1 ;

…n†… 0 †ˆ

dn…t†

dtn

tˆ 0

ˆ

Z 1

1

jnxnfX…x†dxˆjn (^) n:

9

>>>

>>

>>

>>

>>

>>>

=

>>

>>

>>

>>>

>>

>>

>;

… 4 : 50 †

…t†ˆ 1 ‡

X^1

nˆ 1

…jt†n (^) n
n!

:… 4 : 51 †



…t†ˆ… 0 †‡^0 … 0 †t‡^00 … 0 †

t^2

2

‡‡…n†… 0 †

tn

n!

‡...; … 4 : 49 †

 

(^) nˆjn…n†… 0 †;nˆ 1 ; 2 ;...: … 4 : 52 †



pX…k†ˆ

n

k



pk… 1 p†nk;kˆ 0 ; 1 ;...;n:

4.5.1 G eneration of M oments