Digital Signal Processing Reference
In-Depth Information
A simple and very important case is the expected value or mean of a
random variable
X
, which is defined as
∞
E
{
X
}
xp
X
(
x
)
dx
(2.57)
−∞
where
E
denotes the statistical expectation operator. Another important
case is the variance or second central moment, defined as
{·}
∞
2
p
X
(
var
(
X
)
(
x
−
κ
1
(
X
))
x
)
dx
(2.58)
−∞
{
}
where κ
1
(
.
We can generalize the idea of mean and variance by revisiting the notion
of function of an r.v. Let
X
be an r.v. and
f
X
)
=
E
X
(
·
)
an arbitrary function, so that
Y
=
f
(
X
)
(2.59)
It is clear that
Y
is also a random variable and it is possible to define the
expected value thereof as
∞
E
{
Y
}
yp
Y
(
y
)
dy
(2.60)
−∞
or rather
∞
E
f
)
(
X
f
(
x
)
p
X
(
x
)
dx
(2.61)
−∞
Equation 2.61 generalizes the concept of the mean of an r.v. to the expec-
tation of an arbitrary function of the same variable: such procedure will be
of particular relevance throughout the entire topic. At this point, it is worth
pointing out a special case: if we make
f
X
n
in (2.61), we obtain the
n
th
(
X
)
=
moment
of the pdf
p
X
(
x
)
, defined as
∞
x
n
p
X
(
κ
n
(
X
)
=
x
)
dx
(2.62)
∞
From (2.59) and (2.62) it is clear that κ
1
(
.
Even though the moments can be directly computed using (2.62), a
special function can be tailored to produce them directly. The moment-
generating function or first characteristic function of a real r.v.
X
is defined as
X
)
=
E
{
X
}
