Random Variable - Random Signals and Processes Primer with MATLAB - page 68

From ( 2.158 ), we have:

b ¼ y 1 ; a ¼ y 2 b ¼ y 2 y 1 : (2.159)

Placing ( 2.159 ) into ( 2.157 ), we get:

Y ¼ðy 2 y 1 ÞX þ y 1 :

(2.160)

Therefore, the linear transformation ( 2.160 ) of the uniform variable in the
interval [0, 1] results in a uniform random variable in the interval [ y 1 , y 2 ].
Example 2.6.5 The random variable is uniform in the interval [0, 1]. Find the PDF
for the random variable

Y ¼ ln X: (2.161)

Solution The transformation ( 2.161 ) is a monotone transformation (as shown in
Fig. 2.34a ).
From ( 2.161 ), we have:

y ¼ ln x
d y
d x

¼ e y

(2.162)

:

Fig. 2.34 Illustration of the transformation in Example 2.6.5