Random Variable - Random Signals and Processes Primer with MATLAB - page 124

From Fig. 2.57a , we have:

8

0 for x 0 ;

>
<

0 : 5 x for 0 <x 1 ;

F X ðxÞ ¼ 0 : 5 for 1 <x 2 ; (2.456)

0 : 5 ðx 1 Þ for 2 <x 3 ;

:

1 for x> 3 :

The PDF is obtained as the derivative of the distribution ( 2.456 )

8

0 for x 0 ;

>
<

0 : 5 for 0 <x 1 ;

f X ðxÞ ¼ 0 for 1 <x 2 ; (2.457)

0 : 5 for 2 <x 3 ;

:

0 for x> 3 :

The PDF is shown in Fig. 2.57b .
The probability that X is less than 2.5 is equal to the shaded area in Fig. 2.57b :

2

ð

: 5

PfX < 2 : 5 g ¼ f X ðxÞ d x ¼ 0 : 5 þ 0 : 5 0 : 5 ¼ 0 : 75 : (2.458)

0

Exercise E.2.4 The distribution of the random variable is shown in Fig. 2.58 . Find
the type of the random variable and find the probability that the random variable is
equal to 3; also find the probability that the random variable is less than 3.

Answer The random variable is discrete because its distribution is in the form of
step functions with jumps in the discrete values of the random variable: 1, 2, 4,
and 6.5. The length of the jumps is equal to the corresponding probabilities, which
are here equal to 0.25. The corresponding PDF has delta functions in discrete
points, as shown in Fig. 2.58b . The area of each delta functions is equal to 0.25.
The r.v. does not take the value 3 and consequently

PfX ¼ 3 g ¼ 0 : (2.459)

Fig. 2.58 Distribution ( a ) and PDF ( b )