Random Variable - Random Signals and Processes Primer with MATLAB - page 72

Note that for the values of X in the interval [ 1, 1], the transformation is
nonmonotone because two input values correspond to only one output value.
However, for 1 < X < 5, the transformation is monotone,

jj for 1 < x < ;

1

y ¼ (2.183)

x for 1 < x < 5 :

Find the density of the variable Y .

Solution The density of the variable X is given as:

1 = 6 for 1 x 5 ;

f X ðxÞ¼ (2.184)

0 otherwise :

For 1 < x < 1, it follows: 0 < y < 1. Using ( 2.170 ), we have:

f Y ðyÞ¼ 2 f X ðxÞ¼ 1 = 3 for 0 y 1 : (2.185)

For 1 < x < 5, it follows: 1 < y < 5. Using ( 2.140 ), we have:

f Y ðyÞ¼f X ðxÞ¼ 1 = 6 for 1 y 5 : (2.186)

From ( 2.182 ), there is no solution for y <

0 and

f Y ðyÞ¼ 0 for y < 0 : (2.187)

The PDF of the r.v. Y is shown in Fig. 2.36c .

2.6.4 Transformation of Discrete Random Variables

The monotone and nonmonotone transformations considered for the continuous
random variables in Sects. 2.6.2 and 2.6.3 may be easily applied to the
discrete random variables, taking into account that the PDF and distribution for
a discrete random variable are given as:
f X ðxÞ¼ X
i

Pfx i gdðx x i Þ;

F X ðxÞ¼ X
i

Pfx i guðx x i Þ: (2.188)

If the transformation

Y ¼ gðXÞ (2.189)