Biomedical Engineering Reference
In-Depth Information
Jointly Distributed Random Variables
We may consider the case of multiple random variables depending on ! according
to a joint distribution
F
W
1
W
2
:::W
n
P.! W w
1
w
1
;:::;w
n
w
n
/:
In this case, the joint p.d.f. reads
@
n
@x
1
@x
2
:::@x
n
F
W
1
W
2
:::W
n
:
p
W
1
W
2
:::W
n
First order moments read
Z
E
w
j
D
w
j
p
W
1
W
2
:::W
n
d
w
1
d
w
2
:::d
w
n
:
n
R
Second order central moments form the symmetric
covariance
matrix
E
.
w
j
E
w
j
/.
w
k
E
.
w
k
//
jk
Œ
Z
.
w
j
E
w
j
/.
w
k
E
D
.
w
k
//p
W
1
W
2
:::W
n
d
w
1
d
w
2
:::d
w
n
R
n
where clearly
jj
D
j
, the variance of
w
j
.The
correlation coefficient
between
w
j
and
w
k
is defined as
jk
j
k
jk
:
(6.4)
For instance, two jointly distributed Gaussian variables have the distribution
exp
2
ı
T
ƒ
1
ı
;
1
2
p
jƒj
1
p
W
1
W
2
.
w
1
;
w
2
/ D
w
1
1
w
2
2
;ƒD
1
12
12
2
is the covariance matrix, jƒj stands for its
where ı D
determinant, and
1
and
2
are the means of the two variables.
In the sequel, this distribution is denoted by
.;ƒ/. In particular a distribution
with D 0 and ƒ diagonal (that means that the components of the vector are not
correlated) is considered as a model for random disturbances or
white noise
.
3
G
3
The choice of Gaussian distribution for white noise is reasonable, but arbitrary. We could have
considered other distributions for zero-mean, uncorrelated components.
Search WWH ::
Custom Search