Digital Signal Processing Reference
In-Depth Information
From (
4.128
), it follows that
X
is also a normal random variable with parameters:
s
2
m ¼ m
1
m
2
¼
2
;
¼
4
þ
9
¼
13
:
(4.130)
Therefore, the PDF of the variable
X
is given as:
1
26
p
e
x
2
=
26
p
f
x
ðxÞ¼
:
(4.131)
4.6.2
N-Jointly Normal Random Variables
Two jointly normal random variables can be extended to
N
-dimensional case, thus
leading to
N
-jointly normal random variables. From (
4.124
), we can conclude that
the corresponding expression for the
N
-dimensional case would be very complex.
It is much simpler to write the
N
-dimensional density in its matrix form.
To this end, consider the column vector
X
of random variables, such that its
transpose is a (1
N
) row vector given as:
T
X
¼½X
1
;
...
; X
N
;
(4.132)
where upper index T means transpose.
The parameters of the normal vector (
4.132
) are the mean vector
m
X
and the
(
N N
) covariance matrix
C
X
, given in the following expression:
T
T
m
X
¼ E fX
g¼½m
1
;
...
; m
N
;
(4.133)
where
m
i
is the mean value of the random variable
X
i
.
The covariance matrix
C
X
is an (
N N
) matrix of variances and covariances,
2
4
3
5
;
s
1
C
1
;
2
C
1
;N
...
n
o
¼
s
2
C
2
;
1
C
2
;N
...
T
C
X
¼ E
ðX m
X
ÞðX m
X
Þ
(4.134)
...
...
...
...
s
N
C
N;
1
C
N;
2
...
where
s
i
is the variance of the random variable
X
i
, and
C
i,j
is the covariance of
the random variables
X
i
and
X
j
.
Because of
C
i,j
¼ C
j,i
the covariance matrix is
symmetric. Denoting the determinant and the inverse matrix of the matrix (
4.134
)
as |
C
X
| and
C
X
, respectively, we have the PDF of
N
-jointly normal variables:
T
C
X
ðX m
X
Þ
2
ðX m
X
Þ
1
f
X
ðXÞ¼
e
:
(4.135)
p
C
X
N=
2
ð
2
pÞ
j
j
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