Biomedical Engineering Reference
In-Depth Information
where
A
(
) into disjoint intervals, the CDF
F
z
may be
found using the CDF
F
x
. The second approach, called the PDF technique, involves evaluating
γ
)
={
x
:
g
(
x
)
≤
γ
}
. By partitioning
A
(
γ
∞
f
x
(
α
i
(
γ
))
f
z
(
γ
)
=
g
(1)
(
))
,
(6.34)
α
i
(
γ
i
=
1
where
g
−1
(
α
i
=
α
i
(
γ
)
=
γ
)
,
i
=
1
,
2
,...,
(6.35)
denote the distinct solutions to
g
(
. Typically, the PDF technique is much simpler to
use than the CDF technique. However, the PDF technique is applicable only when
z
α
i
)
=
γ
=
g
(
x
)
is continuous and does not equal a constant in any interval in which
f
x
is nonzero.
Next, we evaluated the probability distribution of a random variable
z
y
) created
from jointly distributed random variables
x
and
y
using two approaches. The first approach, a
CDF technique, involves evaluating
=
g
(
x
,
F
z
(
γ
)
=
P
(
z
≤
γ
)
=
P
(
g
(
x
,
y
)
≤
γ
)
=
P
((
x
,
y
)
∈
A
(
γ
))
,
(6.36)
where
A
(
γ
)
={
(
x
,
y
):
g
(
x
,
y
)
≤
γ
}
.
(6.37)
The CDF for the RV
z
can then be found by evaluating the integral
F
z
(
γ
)
=
dF
x
,
y
(
α, β
)
.
(6.38)
A
(
γ
)
The ease of solution here involves transforming
A
(
) into proper limits of integration. We wish
to remind the reader that the special case of a convolution integral is obtained when random
variables
x
and
y
are independent and
z
γ
y
. The second approach involves introducing
an auxiliary random variable and using the PDF technique applied to two functions of two
random variables.
To find the joint probability distribution of random variables
z
=
x
+
y
)
from jointly distributed random variables
x
and
y
, a bivariate CDF as well as a joint PDF
technique were presented. Using the joint PDF technique, the joint PDF for
z
and
w
can be
found as
=
g
(
x
,
y
) and
w
=
h
(
x
,
∞
f
x
,
y
(
α
,β
i
)
i
J
(
i
)
,
f
z
,
w
(
γ,ψ
)
=
(6.39)
α
,β
i
i
=
1
where
(
α
i
(
γ,ψ
)
,β
i
(
γ,ψ
))
,
=
1
,
2
,...,
i
(6.40)
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