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√
R
vv
)
The ter
m (
√
R
uu
/
ρ
u
v
v
v
=
is the conditional mean estimator of
u
from
and
e
√
R
vv
)
(
√
R
uu
/
u
−
ρ
u
v
v
is the error of this estimator. Thus the bivariate pdf
p
(
u
,v
)
factors into a zero-mean Gaussian pdf for the error
e
, with variance
R
uu
(1
−
ρ
u
v
), and
a zero-mean Gaussian pdf for
v
, with variance
R
vv
. The error
e
and
v
are independent.
From
u
=
r
cos
θ
,
v
=
r
sin
θ
,d
u
d
v
=
r
d
r
d
θ
, it is possible to change variables and
obtain the pdf for the polar coordinates (
r
,θ
)
p
r
θ
(
r
,θ
)
=
r
·
p
u
v
(
u
,v
)
|
u
=
r
cos
θ,v
=
r
sin
θ
.
(1.49)
From here it is possible to integrate over
r
to obtain the marginal pdf for
θ
to obtain the marginal pdf for
r
. But this sequence of steps is so clumsy that it is hard
to find formulas in the literature for these marginal pdfs. There is an alternative, which
demonstrates again the power of complex representations.
θ
and over
1.6.2
Complex representation of the bivariate Gaussian distribution
Let's code the real random variables
u
and
v
as
u
v
11
−
x
x
∗
1
2
=
.
(1.50)
j
j
Then the quadratic form
q
u
v
(
u
,v
) in the definition of the bivariate Gaussian distribution
(
1.45
) may be written as
4
x
∗
x
1
R
−
1
zz
11
−
x
x
∗
j
1
,v
=
q
u
v
(
u
)
1
−
j
j
j
x
x
∗
=
x
∗
x
R
−
1
xx
,
(1.51)
where the covariance matrix
R
xx
and its inverse
R
−
1
are
xx
R
xx
E
x
x
∗
x
∗
1
R
zz
11
−
R
xx
x
=
j
R
xx
=
=
,
(1.52)
R
xx
R
xx
1
−
j
j
j
R
xx
.
1
R
−
1
zz
11
−
−
R
xx
1
R
xx
−|
R
xx
|
j
R
−
1
1
4
xx
=
=
(1.53)
−
R
xx
1
−
j
j
j
R
xx
2
The new terms in this representation of the quadratic form
q
u
v
(
u
,v
) bear comment.
So let's consider the elements of
R
xx
. The variance term
R
xx
is
2
R
xx
=
E
|
x
|
=
E
[(
u
+
j
v
)(
u
−
j
v
)]
=
R
uu
+
R
vv
+
j0
.
(1.54)
This variance alone is an incomplete characterization for the bivariate pair (
u
,v
), and it
ρ
u
v
, the correlation coefficient between the random
carries no information at all about
v
variables
u
and
.But
R
xx
contains another complex second-order moment
j2
R
uu
√
R
vv
ρ
u
v
,
R
xx
=
Ex
2
=
E
[(
u
+
j
v
)(
u
+
j
v
)]
=
R
uu
−
R
vv
+
(1.55)
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