Page 118 - Fundamentals of Probability and Statistics for Engineers

Expectations and Moments 101

Another useful expansion is the power series representation of the logarithm
of the characteristic function; that is,

1 n
jt† n
X
log X t†ˆ ; 4:55†
n!
nˆ1
where coefficients n are again obtained from
d n
n
n ˆ j log X t† : 4:56†
dt n
tˆ0

The relations between coefficients n and moments n can be established by
forming the exponential of log X (t), expanding this in a power series of jt, and
equating coefficients to those of corresponding powers in Equation (4.51). We
obtain
1 ˆ 1 ; 9
>
>
>
2
>
2 ˆ 2 ; >
1
=
4:57†
3
3 ˆ 3 3 1 2 ‡ 2 ; >
1 >
>
>
4
2
>
2
4 ˆ 4 3 4 1 3 ‡ 12 2 6 : ;
2 1 1
It is seen that 1 is the mean, 2 is the variance, and 3 is the third central
moment. The higher order n are related to the moments of the same order or
lower, but in a more complex way. Coefficients n are called cumulants of X
and, with a knowledge of these cumulants, we may obtain the moments and
central moments.
4.5.2 INVERSION FORMULAE
Another important use of characteristic functions follows from the inversion
formulae to be developed below.
Consider first a continuous random variable X. We observe that Equation
(4.47) also defines X (t) as the inverse Fourier transform of f (x). The other
X
half of the Fourier transform pair is
1 Z 1 jtx
f x†ˆ e X t†dt: 4:58†
X
2 1
This inversion formula shows that knowledge of the characteristic function
specifies the distribution of X. Furthermore, it follows from the theory of

TLFeBOOK