Mathematical Methods for Physics and Engineering : A Comprehensive Guide

30.6 FUNCTIONS OF RANDOM VARIABLES

functiong(y) for the new random variableY? We now discuss how to obtain

this function.

30.6.1 Discrete random variables

IfXis a discrete RV that takes only the valuesxi,i=1, 2 ,...,n,thenYmust

also be discrete and takes the valuesyi=Y(xi), although some of these values

may be identical. The probability function forYis given by

g(y)=

{∑

jf(xj)ify=yi,

0otherwise,

(30.56)

where the sum extends over those values ofjfor whichyi=Y(xj). The simplest

case arises when the functionY(X) possesses a single-valued inverseX(Y). In this

case, only onex-value corresponds to eachy-value, and we obtain a closed-form

expression forg(y) given by

g(y)=

{

f(x(yi)) ify=yi,

0otherwise.

IfY(X) does not possess a single-valued inverse then the situation is more

complicated and it may not be possible to obtain a closed-form expression for

g(y). Nevertheless, whatever the form ofY(X), one can always use (30.56) to

obtain the numerical values of the probability functiong(y)aty=yi.

30.6.2 Continuous random variables

IfXis a continuous RV, then so too is the new random variableY=Y(X). The

probability thatYlies in the rangeytoy+dyis given by

g(y)dy=

∫

dS

f(x)dx, (30.57)

wheredScorresponds to all values ofxfor whichYlies in the rangeytoy+dy.

Once again the simplest case occurs whenY(X) possesses a single-valued inverse

X(Y). In this case, we may write

g(y)dy=

∣

∣

∣

∣

∫x(y+dy)

x(y)

f(x′)dx′

∣

∣

∣

∣=

∫x(y)+|dxdy|dy

x(y)

f(x′)dx′,

from which we obtain

g(y)=f(x(y))

∣

∣

∣

∣

dx

dy

∣

∣

∣

∣. (30.58)