Random Variable - Random Signals and Processes Primer with MATLAB - page 131

Fig. 2.65 Output ( a ) PDF and ( b ) distribution

Exercise E.2.11 For convenience, the PDF of the variable X , which is the same as
in Exercise E.2.10, is shown again in Fig. 2.66a . The transformation of the random
variable X is shown in Fig. 2.66b . Find the PDF, distribution, and mean value of the
output variable Y .

Answer The output variable Y is a mixed variable with two discrete values, U and
U , and continuous interval [ U , U ].
The variable Y has a discrete value of U , when the input variable is less than
U , resulting in:

5 ð

U

e x d x ¼ 0 5e U PfY ¼Ug¼PfX Ug¼ 0 : : : (2.486)

1

Similarly,

5 1
U

e x d x ¼ 0 5e U PfY ¼ Ug¼PfX Ug¼ 0 : : ¼ PfY ¼Ug: (2.487)

Fig. 2.66 Input ( a ) PDF and ( b ) limiter