Random Variable - Random Signals and Processes Primer with MATLAB - page 70

The relation presented in ( 2.167 ) can be expressed in terms of the corresponding
density functions (see ( 2.64 )), knowing that d x is an infinitesimal value,

Pfx 1 <X x 1 þ d xg¼f X ðx 1 Þ d x;
Pfx 2 d x <X x 2 g¼Pfx 2 <X x 2 þ d xg¼f X ðx 2 Þ d x: (2.168)

From ( 2.167 ) and ( 2.168 ), we have:

f Y ðyÞ d y ¼ f X ðx 1 Þ d x þ f X ðx 2 Þ d x: (2.169)

From here, we arrive at:

f X ðx 1 Þ
d d x ðx 1 Þ

f X ðx 2 Þ
d d x ðx 2 Þ

f Y ðyÞ¼ þ : (2.170)

Next, the right side of ( 2.170 ) is expressed in terms of y , resulting in:

x 1 ¼g 1 ðyÞ
x 2 ¼g 1

f X ðx 1 Þ
d d x ðx 1 Þ

f X ðx 2 Þ
d d x ðx 2 Þ

f Y ðyÞ¼ þ ; (2.171)

ðyÞ

where x 1 and x 2 are the solutions of ( 2.164 ).
For the values of y , for which ( 2.164 ) does not have the real solutions, it follows:

f Y ðyÞ¼ 0 : (2.172)

The results of ( 2.171 ) and ( 2.172 ) can be generalized to include a case in which
( 2.164 ) has N real solutions

x i ¼g 1 ðyÞ;i¼ 1 ;:::;N

f Y ðyÞ¼ X

N

f X ðx i Þ
d y
d x ðx i Þ

: (2.173)

i¼ 1

Example 2.6.6 Find the PDF for the random variable Y , which is obtained by
squaring the random variable X ,

Y ¼ X 2 : (2.174)