Biomedical Engineering Reference
In-Depth Information
PDF
f
x
,
y
, and we wish to evaluate the conditional PDF for random variable
z
y
), given
that event
A
occurred. Depending on whether the event
A
is defined on the range or domain of
z
=
g
(
x
,
y
), one of the following two methods may be used to determine the conditional PDF
of
z
using the bivariate PDF technique.
=
g
(
x
,
i. If
A
is an event defined for an interval on
z
, the conditional PDF,
f
z
|
A
, is computed by first
evaluating
f
z
using the technique in this section. Then, by the definition of a conditional
PDF, we have
)
P
(
A
)
,
f
z
(
γ
f
z
|
A
(
γ
|
A
)
=
γ
∈
A
.
(6.30)
ii. If
A
is an event defined for a region in the
x
y
plane, we will use the conditional PDF
f
x
,
y
|
A
to evaluate the conditional PDF
f
z
|
A
as follows. First, introduce an auxiliary random
variable
w
−
=
h
(
x
,
y
), and evaluate
∞
f
x
,
y
|
A
(
α
i
(
γ,ψ
)
,β
i
(
γ,ψ
)
|
A
)
f
z
,
w
|
A
(
γ,ψ
|
A
)
=
,
(6.31)
|
J
(
α
i
(
γ,ψ
)
,β
i
(
γ,ψ
))
|
i
=
1
where (
α
,β
i
),
i
=
1
,
2
,...
, are the solutions to
γ
=
g
(
α, β
) and
ψ
=
h
(
α, β
) in region
i
A
. Then evaluate the marginal conditional PDF
∞
f
z
|
A
(
γ
|
A
)
=
f
z
,
w
|
A
(
γ,ψ
|
A
)
d
ψ.
(6.32)
−∞
Example 6.4.7.
Random variables x and y have joint PDF
3
α,
0
<β<α<
1
f
x
,
y
(
α, β
)
=
0
,
otherwise
.
Find the PDF for z
=
x
+
y , given event A
={
max(
x
,
y
)
≤
0
.
5
}
.
Solution.
We begin by finding the conditional PDF for
x
and
y
, given
A
. From Figure 6.11(a)
we find
α
0
.
5
1
8
;
P
(
A
)
=
f
x
,
y
(
α, β
)
d
α
d
β
=
3
α
d
β
d
α
=
A
0
0
consequently,
⎨
f
x
,
y
(
α, β
)
=
24
α,
0
<β<α<
0
.
5
f
x
,
y
|
A
(
α, β
|
A
)
=
P
(
A
)
⎩
0
,
otherwise
.
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