Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.6 FUNCTIONS OF RANDOM VARIABLES

where the range of integration is over all possible values of the variablesxi.This

integral is most readily evaluated by substituting in (30.62) the Fourier integral

representation of the Dirac delta function discussed in subsection 13.1.4, namely

δ(Z(x 1 ,x 2 ,...,xn)−z)=

1

2 π

∫∞

−∞

eik(Z(x^1 ,x^2 ,...,xn)−z)dk. (30.63)

This is best illustrated by considering a specific example.

A general one-dimensional random walk consists ofnindependent steps, each of which

can be of a different length and in either direction along thex-axis. Ifg(x)is the PDF for

the (positive or negative) displacementXalong thex-axis achieved in a single step, obtain

an expression for the PDF of the total displacementSafternsteps.

The total displacementSis simply the algebraic sum of the displacementsXiachieved in
each of thensteps, so that

S=X 1 +X 2 +···+Xn.

Since the random variablesXiare independent and have the same PDFg(x), their joint
PDF is simplyg(x 1 )g(x 2 )···g(xn). Substituting this into (30.62), together with (30.63), we
obtain

p(s)=

∫∞

−∞

∫∞

−∞

···

∫∞

−∞

g(x 1 )g(x 2 )···g(xn)

1

2 π

∫∞

−∞

eik[(x^1 +x^2 +···+xn)−s]dk dx 1 dx 2 ···dxn

=

1

2 π

∫∞

−∞

dk e−iks

(∫∞

−∞

g(x)eikxdx

)n

. (30.64)

It is convenient to define thecharacteristic functionC(k) of the variableXas

C(k)=

∫∞

−∞

g(x)eikxdx,

which is simply related to the Fourier transform ofg(x). Then (30.64) may be written as

p(s)=

1

2 π

∫∞

−∞

e−iks[C(k)]ndk.

Thusp(s) can be found by evaluating two Fourier integrals. Characteristic functions will
be discussed in more detail in subsection 30.7.3.

30.6.4 Expectation values and variances

In some cases, one is interested only in the expectation value or the variance

of the new variableZrather than in its full probability density function. For

definiteness, let us consider the random variableZ=Z(X, Y), which is a function

of two RVsXandYwith a known joint distributionf(x, y); the results we will

obtain are readily generalised to more (or fewer) variables.

It is clear thatE[Z]andV[Z] can be obtained, in principle, by first using the

methods discussed above to obtainp(z) and then evaluating the appropriate sums

or integrals. The intermediate step of calculatingp(z) is not necessary, however,

since it is straightforward to obtain expressions forE[Z]andV[Z] in terms of