Geology Reference
In-Depth Information
2
The
i
th spectral estimate entering the product is χ
ν
i
distributed with probability
density function
1
(ν
i
/2)
x
ν
i
/2
−
1
p
i
(
x
i
)
=
exp (
−
x
i
/2).
(2.396)
i
2
ν
i
/2
Γ
On the assumption that the individual spectral estimates are statistically independ-
ent, the cumulative distribution function for the spectral estimates in the product
spectrum is
∞
p
1
(
x
1
)
∞
0
∞
p
n
−
1
(
x
n
−
1
)
∞
0
F
(
z
)
=
p
2
(
x
2
)
···
p
n
(
x
n
)
0
0
z
ξ
n
1
(ν
n
+
1
/2)
x
ν
n
+
1
/2
−
1
·
exp(
−
x
n
+
1
/2)
dx
n
+
1
dx
n
···
dx
1
.
n
+
1
2
ν
n
+
1
/2
Γ
0
(2.397)
With the change of the variable of integration to
t
=
x
n
+
1
/2, the innermost integral
becomes
z
ξ
n
/2
1
t
ν
n
+
1
/2
−
1
e
−
t
dt
=
P
(ν
n
+
1
/2,
z
ξ
n
/2),
(2.398)
Γ
(ν
n
+
1
/2)
0
where
y
1
t
a
−
1
e
−
t
dt
P
(
a
,y)
=
(2.399)
Γ
(
a
)
0
is the incomplete gamma function and
Γ
(
a
) is the gamma function. The gamma
function has the integral representation
∞
t
a
−
1
e
−
t
dt
.
Γ
(
a
)
=
(2.400)
0
Calculation of the incomplete gamma function then requires calculation of the
gamma function and the integral
y
t
a
−
1
e
−
t
dt
.
γ(
a
,y)
=
(2.401)
0
As a function of y the incomplete gamma function ranges from
P
(
a
,0)
=
0to
P
(
a
,
∞
)
=
1. Its complement is then
∞
y
1
1
t
a
−
1
e
−
t
dt
t
a
−
1
e
−
t
dt
1
−
P
(
a
,y)
=
−
Γ
(
a
)
Γ
(
a
)
0
0
∞
1
t
a
−
1
e
−
t
dt
=
Γ
(
a
,y)
Γ
=
(
a
)
,
(2.402)
Γ
(
a
)
y
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