Random Variable - Random Signals and Processes Primer with MATLAB - page 137

for three cases (Fig. 2.70b ):

(a) X is uniform in the interval [ 2, 1]
(b) X is uniform in the interval [0, 2]
(c) X is uniform in the interval [2, 4],
Answer
(a) The random variable Y is a mixed variable that has the discrete value y ¼ 0, and
it is continuous in the range [0, 1].

PfY ¼ 0 g¼PfX < 0 g¼ 2 = 3 : (2.516)

For 0 < x < 1, the output random variable Y is continuous and equal to X ,

resulting in:

f Y ðyÞ¼f X ðxÞ¼ 1 = 3 : (2.517)

From ( 2.516 ) and ( 2.517 ), we arrive at:

2 = 3 dðyÞþ 1 = 30 y 1 ;

f Y ðyÞ¼ (2.518)

0 otherwise :

(b) The random variables X and Y are both positive and, from ( 2.515 ), the random
variable is equal to X in the interval [0, 2].

=

20

y 2 ;

1

f Y ðyÞ¼

(2.519)

:

0 otherwise

(c) The random variable Y is equal to X in the interval [2, 4]

Y ¼ X for 2 x 4 (2.520)

and

1 = 22 y 4 ;

f Y ðyÞ¼ (2.521)

0 otherwise :

Exercise E.2.18 The discrete random variable X has discrete values 1, 2, 3, 6,
and 8, with the corresponding probabilities:

PfX ¼ 1 g¼PfX ¼ 8 g¼ 0 : 1 ;

PfX ¼ 2 g¼PfX ¼ 3 g¼ 0 : 3 ;

PfX ¼ 6 g¼ 0 : 2 : (2.522)