Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

so thatCX(t)=MX(it), whereMX(t)istheMGFofX. Clearly, the characteristic

function and the MGF are very closely related and can be used interchangeably.

Because of the formal similarity between the definitions ofCX(t)andMX(t), the

characteristic function possesses analogous properties to those listed in the previ-

ous section for the MGF, with only minor modifications. Indeed, by substitutingit

fortin any of the relations obeyed by the MGF and noting thatCX(t)=MX(it),

we obtain the corresponding relationship for the characteristic function. Thus, for

example, the moments ofXare given in terms of the derivatives ofCX(t)by

E[Xn]=(−i)nC

(n)

X(0).

Similarly, ifY=aX+bthenCY(t)=eibtCX(at).

Whether to describe a random variable by its characteristic function or by its

MGF is partly a matter of personal preference. However, the use of the CF does

have some advantages. Most importantly, the replacement of the exponentialetX

in the definition of the MGF by the complex oscillatory functioneitXin the CF

means that in the latter we avoid any difficulties associated with convergence of

the relevant sum or integral. Furthermore, whenXis a continous RV, we see

from (30.91) thatCX(t) is related to the Fourier transform of the PDFf(x). As

a consequence of Fourier’s inversion theorem, we may obtainf(x)fromCX(t)by

performing the inverse transform

f(x)=

1

2 π

∫∞

−∞

CX(t)e−itxdt.

30.7.4 Cumulant generating function

As mentioned at the end of subsection 30.5.5, we may also describe a probability

density functionf(x) in terms of itscumulants. These quantities may be expressed

in terms of the moments of the distribution and are important in sampling theory,

which we discuss in the next chapter. The cumulants of a distribution are best

defined in terms of its cumulant generating function (CGF), given byKX(t)=

lnMX(t)whereMX(t) is the MGF of the distribution. IfKX(t)isexpandedasa

power series intthen thekth cumulantκkoff(x) is the coefficient oftk/k!:

KX(t)=lnMX(t)≡κ 1 t+κ 2

t^2

2!

+κ 3

t^3

3!

+···. (30.92)

SinceMX(0) = 1,KX(t) contains no constant term.

Find all the cumulants of the Gaussian distribution discussed in the previous example.

The moment generating function for the Gaussian distribution isMX(t)=exp

(

μt+^12 σ^2 t^2

)

.

Thus, the cumulant generating function has the simple form

KX(t)=lnMX(t)=μt+^12 σ^2 t^2.