Random Variable - Random Signals and Processes Primer with MATLAB - page 94

The third integral in ( 2.299 ) is:

ð

2

1
2 þ

2
1 ¼

2 x 3
9

2 x 2
3

2 xðx 2 Þ
3

4
9 :

d x ¼ (2.303)

1

From ( 2.299 ), and using ( 2.301 )-( 2.303 ), we arrive at:

EfXg ¼ 2 =

9 þ 4

=

9 ¼ 2

= :

3 (2.304)

2.7.5 Conditional Mean Values

There are some applications where we need to compute mean value conditioned on
certain events.
The mean value of the random variable X , conditioned on a given event A , can be
calculated using the general expression ( 2.261 ) by replacing the PDF with condi-
tional PDF as shown in ( 2.305 ):

1

E XjA f g ¼ xf X ðxjAÞ d x: (2.305)

1

Similarly, for the function g ( X ) of the random variable X , conditioned on event
A , the corresponding mean value is given as

1

E gðXÞjA f g ¼ gðxÞf X ðxjAÞ d x: (2.306)

1

The next issue is how to define event A .
One way to define event A is to let it be dependent on the random variable X , for
example,

A ¼ fXag: (2.307)

Then the corresponding conditional density is (see ( 2.115 )):

f X ðxÞ
PfXAg ¼

f X ðxÞ

f X ðxjAÞ ¼ f X ðxjXaÞ ¼

f X ðxÞ d x :

(2.308)

Ð

a

1