Biomedical Engineering Reference
In-Depth Information
y
ψ
γ
x
γ
−
FIGURE 6.7:
Integration region for Example 6.4.1.
and
h
−1
((
−∞
,ψ
])
={
(
x
,
y
):
y
≤
ψ
}
.
The intersection of these regions is
A
(
γ,ψ
)
={
(
x
,
y
):
y
≤
min(
γ
−
x
,ψ
)
}
,
which is illustrated in Figure. 6.7. With the aid of Figure 6.7 and (6.20) we find
ψ
γ
−
β
F
z
,
w
(
γ,ψ
)
=
dF
x
,
y
(
α, β
)
.
−∞
−∞
Instead of carrying out the above integration and then differentiating the result, we differentiate
the above integral to obtain the PDF
f
z
,
w
directly. We find
ψ
γ
−
β
∂
F
z
,
w
(
γ,ψ
)
1
h
1
=
lim
h
1
dF
x
,
y
(
α, β
)
,
∂γ
→
0
−∞
γ
−
h
1
−
β
and
ψ
γ
−
β
2
F
z
,
w
(
∂
γ,ψ
)
1
h
1
h
2
=
lim
h
2
lim
h
1
dF
x
,
y
(
α, β
)
.
∂ψ ∂γ
→
0
→
0
ψ
−
h
2
γ
−
h
1
−
β
Performing the indicated limits, we find that
f
z
,
w
(
γ,ψ
)
=
f
x
,
y
(
γ
−
ψ, ψ
); substituting, we
obtain
0
.
25
,
0
<γ
−
ψ<
2
,
0
<ψ<
2
f
z
,
w
(
γ,ψ
)
=
0
,
otherwise
.
When the RVs
x
and
y
are jointly continuous, it is usually easier to find the joint PDF
f
z
,
w
than to carry out the integral indicated in (6.20).
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