Random Variable - Random Signals and Processes Primer with MATLAB - page 93

Similarly, the second integral in ( 2.294 ) is:

1

xdðx 1 Þ d x ¼ 1 (2.296)

1

From ( 2.294 )to( 2.296 ), we have:

m X ¼ 0 : 5 0 þ 0 : 5 1 ¼ 0 : 5 : (2.297)

Note that this is the same result as in Example 2.7.2
The next example illustrates that the general expression ( 2.261 ) can also be used
in the calculation of the mean value for a mixed random variable.
Example 2.7.14 Consider a mixed random variable with the density function:

8
<

1
3 dðxÞ

for x ¼ 0 ;

2
3 x

for 0 < x 1 ;

f X ðxÞ¼ (2.298)

:

2
3 ðx 2 Þ

for 1 x 2 ;

0 otherwise :

From ( 2.261 ) and ( 2.298 ), we have:

1 1 ð ð

1 2

x
3 dðxÞ d x þ

2 x 2
3 d x þ

2 x ð x 2 Þ
3

EfXg¼ xf X ðxÞ d x ¼ d x: (2.299)

1 1 0 1

Knowing that

1

xdðxÞ d x ¼ 0 (2.300)

1

the first integral in the right side of ( 2.299 ) is equal to zero, that is:

1

x
3 dðxÞ d x ¼ 0

(2.301)

1

The second integral on the right side of ( 2.299 ) is:

ð

1

1
0 ¼

2 x 2
3 d x ¼

2 x 3
9

2
9 :

(2.302)

0