Geoscience Reference
In-Depth Information
The decision-making system is synthesized according to the principal scheme of
Fig.
3.5
. Its functioning scheme is shown in Fig.
3.7
.
3.4 Important Parameters of the Sequential Analysis
Procedure
An investigation of the sum of x
n
of independent random values having the same
distribution imposes a double task on the distribution function, P(x
n
< x)=F
1
(x),
both for
fixed n and for the variable case. The situation of comparison between x
n
and some level C arises in both cases. However, in the second case this task is
transformed into the study of the distribution function, P(
m
< n)=F
2
(n), of the
chance number
m
of components by which x
m
first exceeds the level C = C(
α
,
ʲ
):
x
i
< C (i =1,
1), x
m
C.
In conformity with the central limit theorem, in the
…
,
m −
first case the distribution
→
∞
F
1
(x) approaches the normal distribution when n
. In the second case, we have
the distribution represented by expression (
3.12
). The following correlation
between these distributions exists:
h
i
þ U
x
þ
1
h
i
exp 2
fg;
1
=
2
1
=
2
W
c
ðÞ
¼
U
ð
x
1
Þ
c
ðÞ
=
ð
Þ
c
ðÞ
=
ð
3
:
13
Þ
where
is the normal distribution function.
The expression (
3.13
) makes it possible to study the sequential analysis distri-
bution using the characteristics of
U
U
. As seen from (
3.11
), the distribution function
W
c
(x)isde
ned on the half-space [0,
∞
] and it has one maximum at the point
x = m
c
. In fact, we have:
g
¼ 0
1
=
2
exp
Þ
c
1
=
2
=
½y
3
=
2
5cy
þ
y
1
dW
c
ðÞ=
dy ¼ d
ð
=
dy
ð
2
pÞ
½
0
:
2
After the set of transformations, this equation is solved to give (Fig.
3.8
):
h
i
1
=
2
9
þ
4c
2
y ¼ m
c
¼
3
=
2
ðÞ
3
:
14
Þ
The position of the W
c
(y) maximum changes depending only on the parameter
c remaining less than one (m
c
< 1). Moreover, m
c
→
0 when c
0 and m
c
→
1
→
when c
→
∞
. Comparing Eqs. (
3.1
) and (
3.14
), we
nd:
h
i
exp
31
m
c
1
=
2
1
=
2
m
c
1
m
c
W
c
m
ðÞ
¼
3
=ð
2
pÞ
Þ
=
ð
Þ=
½
21
þ
m
c
ð
Þ
1
f
g
ð
3
:
15
Þ
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