Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.6 FUNCTIONS OF RANDOM VARIABLES

y

y+dy

dx 1 dx 2 X

Y

Figure 30.9 Illustration of a functionY(X) whose inverseX(Y) is a double-

valued function ofY. The rangeytoy+dycorresponds toXbeing either in

the rangex 1 tox 1 +dx 1 or in the rangex 2 tox 2 +dx 2.

This result may be generalised straightforwardly to the case where the rangeyto

y+dycorresponds to more than twox-intervals.

The random variableXis Gaussian distributed (see subsection 30.9.1) with meanμand

varianceσ^2. Find the PDF of the new variableY=(X−μ)^2 /σ^2.

It is clear thatX(Y) is a double-valued function ofY. However, in this case, it is
straightforward to obtain single-valued functions giving the two values ofxthat correspond
to a given value ofy;thesearex 1 =μ−σ

√

yandx 2 =μ+σ

√

y,where

√

yis taken to
mean the positive square root.
The PDF ofXis given by

f(x)=

1

σ

√

2 π

exp

[

−

(x−μ)^2

2 σ^2

]

.

Sincedx 1 /dy=−σ/(2

√

y)anddx 2 /dy=σ/(2

√

y), from (30.59) we obtain

g(y)=

1

σ

√

2 π

exp(−^12 y)

∣

∣∣

∣

−σ

2

√

y

∣

∣∣

∣+

1

σ

√

2 π

exp(−^12 y)

∣

∣∣

∣

σ

2

√

y

∣

∣∣

∣

=

1

2

√

π

(^12 y)−^1 /^2 exp(− 21 y).

As we shall see in subsection 30.9.3,this is the gamma distributionγ(^12 ,^12 ).

30.6.3 Functions of several random variables

We may extend our discussion further, to the case in which the new random

variable is a function ofseveralother random variables. For definiteness, let us

consider the random variableZ=Z(X, Y), which is a function of two other

RVsXandY. Given that these variables are described by the joint probability

density functionf(x, y), we wish to find the probability density functionp(z)of

the variableZ.