Random Variable - Random Signals and Processes Primer with MATLAB - page 135

Exercise E.2.15 The random variable X is uniform in the interval [ p , p ]. Find the
PDF of the variable Y if

Y ¼ sin X: (2.507)

Answer The PDF f X ( x ) is shown in Fig. 2.69a .
The random variable Y is a “sin function” and, consequently, takes only the
values in the interval [ 1, 1], resulting in:

f Y ðyÞ¼ 0 j > :

for 1 (2.508)

For | y |

<

1, there are two solutions x 1 and x 2 for

y ¼ sin x; (2.509)

as shown in Fig. 2.68b . From ( 2.509 ), it follows:

¼ cos x

p

1 sin 2 x

p

1 y 2

d y
d x

j j ¼ ¼ : (2.510)

Finally, from ( 2.170 ) and ( 2.510 ), the PDF is given as

x¼x 1 x¼x 2

f X ð x Þ
d d x

f X ð x Þ
d d x

1
2 p

p

f Y ðyÞ¼ þ ¼ 2

1 y 2

1
p

¼ p for 1 y 1 : (2.511)

1 y 2

For other values of y , the PDF is zero.

Fig. 2.69 Input ( a ) PDF and the ( b ) transformation