Digital Signal Processing Reference
In-Depth Information
defines a new random vector on
X
−1
(
U
)
,
A
with
σ
-algebra
A
:=
{
X
−1
(
U
)
A
∈
A
|
A
⊂
}
and probability measure
P
(
A
)
P
X
(
U
)
P
(
A
)=
A
. It is called the
restriction
of
X
to
U
.
for
A
∈
Lemma 3.5 Transformation properties of restriction:
Let
X
,
Y
:Ω
be random variables with densities
p
X
and
p
Y
respec-
tively, and let
U
→ R
n
with
P
X
(
U
)
,P
Y
(
U
)
>
0.
⊂ R
i.
(
λ
X
)
|
(
λU
)=
λ
X
|
U
if
λ
∈ R
.
ii.
(
AX
)
|
(
A
U
)=
A
(
X
|
U
)if
A
∈
Gl(
n
).
iii.
If
X
is independent and
U
=
a
1
,b
1
]
×
...
×
[
a
n
,b
n
], then
X
|
U
is
independent.
We can construct samples of
X
|
U
given samples
x
1
,...,
x
s
of
X
by
taking all samples that lie in
U
.
Examples of Probability Distributions
In this section, we give some important examples of random vectors. In
particular, Gaussian distributed random vectors will play a key role in
ICA. The probability density functions of the following random vectors
in the one-dimensional case are plotted in figure 3.4.
Uniform Density
For a subset
K
n
let
χ
K
⊂ R
denote the
characteristic function
of
K
:
n
χ
K
:
R
−→ R
1
x
∈
K
x
−→
0
x
/
∈
K
Let
K
⊂ R
n
, be a measurable set. A random vector
Definition 3.11:
X
:Ω
n
is said to be uniform in
K
if its density function
p
X
exists
and is of the form
→ R
1
vol(
K
)
χ
K
p
X
=
.
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