Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PROBABILITY

Scaling and shifting

IfY=aX+b,whereaandbare arbitrary constants, then

MY(t)=E

[

etY

]

=E

[

et(aX+b)

]

=ebtE

[

eatX

]

=ebtMX(at). (30.88)

This result is often useful for obtaining thecentralmoments of a distribution. If the

MFG ofXisMX(t) then the variableY=X−μhas the MGFMY(t)=e−μtMX(t),

which clearly generates the central moments ofX,i.e.

E[(X−μ)n]=E[Yn]=M(Yn)(0) =

(

dn

dtn

[e−μtMX(t)]

)

t=0

.

Sums of random variables

IfX 1 ,X 2 ,...,XNare independent random variables andSN=X 1 +X 2 +···+XN

then

MSN(t)=E

[

etSN

]

=E

[

et(X^1 +X^2 +···+XN)

]

=E

[N

∏

i=1

etXi

]

.

Since theXiareindependent,

MSN(t)=

∏N

i=1

E

[

etXi

]

=

∏N

i=1

MXi(t). (30.89)

In words, the MGF of the sum ofNindependent random variables is the product

of their individual MGFs. By combining (30.89) with (30.88), we obtain the more

general result that the MGF ofSN=c 1 X 1 +c 2 X 2 +···+cNXN(where theciare

constants) is given by

MSN(t)=

∏N

i=1

MXi(cit). (30.90)

Variable-length sums of random variables

Let us consider the sum ofNindependent random variablesXi(i=1, 2 ,...,N), all

with the same probability distribution, and let us suppose thatNis itself a random

variable with a known distribution. Following the notation of section 30.7.1,

SN=X 1 +X 2 +···+XN,

whereNis a random variable with Pr(N=n)=hnand probability generating

functionχN(t)=

∑

hntn. For definiteness, let us assume that theXiare continuous

RVs (an analogous discussion can be given in the discrete case). Thus, the