Digital Signal Processing Reference
In-Depth Information
the density
|
−1
p
X
.
p
h
◦
X
◦
h
=
|
det
D
h
Expectation and moments
Definition 3.5: Let
X
be a random vector on a probability space
(Ω
,
A
,P
). If
X
is
P
-integrable (
X
∈
L
1
(Ω
,
R
n
)), then
E
(
X
):=
X
dP
Ω
is called the
expectation of
X
.
E
(
X
) is also called the
mean
of
X
or the
first-order moment
.
∈
L
1
(Ω
,
n
)then
E
(
X
)=
R
Lemma 3.1:
If
X
x
dP
X
.
R
n
Hence
E
(
X
)isa
probability theoretic notion
(i.e. it depends only on
the distribution
P
X
of
X
). If
X
has a density
p
X
,then
E
(
X
)=
R
n
x
p
X
(
x
)
d
x
.
The expectation is a linear mapping of the vector space
L
1
(Ω
,
n
)to
R
n
,so
E
(
AX
)=
AE
(
X
) for a matrix
A
.
R
n
be an
L
2
random vector. Then
Definition 3.6:
Let
X
:Ω
→ R
R
X
:= Cor(
X
):=
E
(
XX
)
C
X
:= Cov(
X
):=
E
((
X
E
(
X
))
)
−
E
(
X
))(
X
−
exist, and are called the
correlation
(respectively
covariance
)of
X
.
Note that
X
is then also
L
1
(i.e. integrable) and therefore
E
(
X
)ex-
ists.
R
X
and
C
X
are symmetric and positive semidefinite (i.e.
a
R
X
a
≥
n
). If
X
has no
deterministic
component (i.e. a component
with constant image), then the two matrices are positive-definite, mean-
ing that
a
R
X
a
>
0for
a
0 for all
a
∈ R
= 0. Since the above equations are quadratic
in
X
, the components of
R
are called the
second-order moments
of
X
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