Graphics Programs Reference
In-Depth Information
F
X
x
()
F
X
x
()
=
Pr x
1
{
≤≤
X
}
(13.60)
2
The
cdf
has the following properties:
0
≤
F
X
() 1
≤
F
X
=
F
X
() 1
(
∞
)
0
(13.61)
=
F
X
x
()
F
X
≤
x
()
⇔
x
1
≤
x
2
It is often practical to describe a random variable by the derivative of its
cdf
,
which is called the Probability Density Function
(pdf).
The
pdf
of the random
variable
X
is
d
F
X
f
X
()
=
()
d
x
(13.62)
or, equivalently,
x
∫
F
X
()
Pr X
=
{
≤
x
}
=
f
X
()λ
d
(13.63)
∞
The probability that a random variable
X
has values in the interval
(
x
1
,
x
2
)
is
x
2
∫
F
X
x
()
F
X
x
()
=
Pr x
1
{
≤≤
X
}
=
f
X
()
d
(13.64)
2
x
1
Define the
nth
moment for the random variable
X
as
∞
∫
EX
[]
X
n
x
n
f
X
==
()
d
(13.65)
∞
The first moment, , is called the mean value, while the second moment,
, is called the mean squared value. When the random variable
E
[]
EX
[]
X
represents an electrical signal across a
1Ω
resistor, then
E
[]
is the DC com-
EX
[]
ponent, and
is the total average power.
The
nth
central moment is defined as
∞
∫
)
n
)
n
)
n
f
X
EX X
[
(
]
=
(
XX
=
(
xx
()
x
d
(13.66)
∞
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