Digital Signal Processing Reference
In-Depth Information
with
p
Y
(
y
0
)
=0,the
conditional density
of
X
with respect to
Y
is the
function
y
0
)=
p
X
,
Y
(
x
,
y
0
)
p
Y
(
y
0
)
p
X
|
Y
(
x
|
n
.
for
x
∈ R
Indeed, it is possible to define a conditional random vector
X
|
Y
with
density
p
X
|
Y
(
x
y
0
).
Note that if
X
and
Y
are independent, meaning that their joint
density factorizes, then
p
X
|
Y
(
x
|
|
y
0
)=
p
X
. More generally we get
p
X
|
Y
(
x
0
|
y
0
)=
p
X
|
Y
(
x
0
|
y
0
)
p
Y
(
y
0
)=
p
Y
|
X
(
y
0
|
x
0
)
p
X
(
x
0
)
,
so we have shown
Bayes's rule
:
x
0
)=
p
X
|
Y
(
x
0
|
y
0
)
p
Y
(
y
0
)
p
X
(
x
0
)
p
Y
|
X
(
y
0
|
Operations on Random Vectors
In this section we present two different methods for constructing new
random vectors out of given ones in order to get certain properties. The
first of these properties is the vanishing mean.
n
is called
centered
if
Definition 3.10:
A random vector
X
:Ω
→ R
E
(
X
)=0.
n
be a random vector. Then
X
Lemma 3.3:
Let
X
:Ω
→ R
−
E
(
X
)is
centered.
Proof
E
(
X
−
E
(
X
)) =
E
(
X
)
−
E
(
X
)=0
.
Another construction we want to make is the restriction of a random
vector in the sense that only samples from a given region are taken into
account. This notion is formalized in next lemma 3.4.
n
be a random vector, and let
U
n
be
Lemma 3.4:
Let
X
:Ω
→ R
⊂ R
measurable with
P
X
(
U
)=
P
(
X
−1
(
U
))
>
0. Then
U
:
X
−1
(
U
)
n
X
|
−→ R
ω
−→
X
(
ω
)
Search WWH ::
Custom Search