Random Variable - Random Signals and Processes Primer with MATLAB - page 69

Finally, from ( 2.162 ) and ( 2.140 ), we obtain the desired PDF:

e y for y 0 ;

f Y ðyÞ¼

(2.163)

:

0 otherwise

which is shown in Fig. 2.34b .

2.6.3 Nonmonotone Transformation

In the case of the nonmonotone transformation,

Y ¼ gðXÞ; (2.164)

more than one value of the input random variable corresponds to only one value of
the output random variable. Consider the simplest case, in which two input values
correspond to only one output value. In the case of continuous random variables,
which means that the output random variable Y is in the interval

ðy; y þ dyÞ; (2.165)

if the input random variable is either in the interval (d x 1 ¼ d x 2 ¼ d x ):

ðx 1 ; x 1 þ d xÞ or ðx 2 d x; x 2 Þ;

(2.166)

as shown in Fig. 2.35 .
Since the events ( 2.166 ) are exclusive, according to Axiom III ( 1.13 ), we can
write:

Pfy <Y y þ d yg¼Pfx 1 <X x 1 þ d xgþPfx 2 d x <X x 2 g: (2.167)

Fig. 2.35 Nonmonotone transformation