Databases Reference
In-Depth Information
Example A.3.2:
For our television scenario,
⎧
⎨
⎩
0
.
4
if
X
=
0
0
.
4
if
X
=
1
0
.
002
if
X
=
2
0
.
02
if
X
=
3
0
.
09
if
X
=
4
f
X
(
x
)
=
0
.
02
if
X
=
5
0
.
04
if
X
=
6
0
.
018
if
X
=
7
0
.
01
if
X
=
8
0
otherwise
Example A.3.3:
For our speech example, the
pdf
is given by
1
2
e
−
2
|
x
|
f
X
(
)
=
x
A.4 Expectation
When dealing with random processes, we often deal with average quantities, like the signal
power and noise power in communication systems, and the mean time between failures in
various design problems. To obtain these average quantities, we use something called an
expectation operator
. Formally, the expectation operator
E
[]
is defined as follows: The
expected value
of a random variable
X
is given by
i
x
i
P
E
[
X
]=
(
X
=
x
i
)
(A.11)
when
X
is a discrete random variable with realizations
{
x
i
}
and by
∞
E
[
X
]=
xf
X
(
x
)
dx
(A.12)
−∞
where
f
X
(
is the
pdf
of
X
.
The expected value is very much like the average value and, if the frequency of occurrence
is an accurate estimate of the probability, is identical to the average value. Consider the
following example:
x
)
