Biomedical Engineering Reference
In-Depth Information
It follows that
x
−
η
x
E
(
y
|
x
)
=
η
+
ρσ
(5.97)
y
y
σ
x
and
2
y
2
2
)
σ
=
σ
y
(1
−
ρ
.
(5.98)
|
x
By interchanging
x
with
y
and
α
with
β
,
⎛
⎞
2
x
β
−
η
y
α
−
η
−
ρσ
x
σ
⎝
−
⎠
y
exp
2(1
−
ρ
2
)
σ
2
y
f
x
|
y
(
α
|
β
)
=
2
,
(5.99)
x
(1
2
)
πσ
−
ρ
y
−
η
y
E
(
x
|
y
)
=
η
+
ρσ
,
(5.100)
x
x
σ
y
and
2
x
2
2
)
σ
=
σ
x
(1
−
ρ
.
(5.101)
|
y
A three-dimensional plot of a bivariate Gaussian PDF is shown in Figure 5.11.
The bivariate characteristic function for
x
and
y
is easily obtained as follows. Since
x
and
y
are jointly Gaussian, the RV
t
1
x
+
t
2
y
is a univariate Gaussian RV:
G
t
1
y
.
t
1
σ
2
x
t
2
σ
2
t
1
x
+
t
2
y
∼
η
+
t
2
η
,
+
2
t
1
t
2
σ
+
x
y
x
,
y
f
(
α
,
β
)
x,y
.265
β
=
3
α
=−
3
β
=−
3
α
=
3
FIGURE 5.11:
Bivariate Gaussian PDF
f
x
,
y
(
α, β
) with
σ
=
σ
=
1,
η
=
η
=
0, and
ρ
=−
0
.
8.
x
y
x
y
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