Random Variable - Random Signals and Processes Primer with MATLAB - page 67

Example 2.6.3 Using the result from Example 2.6.1, find the PDF for the random
variable

Y ¼ 0 : 5 X þ 2 ; (2.153)

if X is a uniform random variable in [ 1, 1].

Solution From ( 2.153 ), we have:

y 1 ¼ 0 : 5 ð 1 Þþ 2 ¼ 2 : 5 ; (2.154)

y 2 ¼ 0 : 5 þ 2 ¼ 1 : 5 : (2.155)

Using ( 2.154 ), ( 2.155 ), and ( 2.148 ), we arrive at:

8
<

1
y 1 y 2 ¼ 1

for 1 : 5 y 2 : 5 ;

f Y ðyÞ¼ (2.156)

:

0 otherwise :

The input and output PDFs are shown in Fig. 2.33 .

Example 2.6.4 The random variable X is uniform in the interval [0, 1]. Find the
transformation which will result in an uniform random variable in the interval [ y 1 , y 2 ].
Solution From Example 2.6.1, it follows that the transformation is a linear trans-
formation

Y ¼ aX þ b; (2.157)

where X is the given uniform random variable in the interval [0, 1], and the variable
Y is uniform in the interval [ y 1 , y 2 ].
From ( 2.157 ), using x 1 ¼ 0 and x 2 ¼ 1, we get:

y 1 ¼ ax 1 þ b ¼ b;
y 2 ¼ ax 2 þ b ¼ a þ b:

(2.158)

Fig. 2.33 Illustration of the linear transformation of the uniform random variable in Example
2.6.3