Environmental Engineering Reference
In-Depth Information
{
}
{
}
t
≤
min ...,
t
,... ,
t
≤
max ...,
t
,... ;
min
i
max
i
1
1
- split the operating time interval [
t
min
,
t
max
] into
k
intervals of equal
width ∆
t
(the step of the histogram)
ξ
D= D= − = −
t
t
,
tt t tt
;
i
+
1
i
i
i
−
1
k
- to calculate the frequency of occurrence of failures in all
k
intervals
D
nt t t
(,
+D
)
D
nt t
(,
)
ˆ
P
=
i i
=
i i
+
1
,
i
N
N
+D
is the number of objects that fail in the interval
[
]
where
D
nt t t
(,
)
tt t
,
+D
.
i i
i i
It is obvious that
k
ˆ
∑
P
=
1.
1
The resultant statistical series is represented as a histogram which is
constructed as follows. The intervals ∆
t
are plotted on the abscissa and
each interval is used as a base for constructing a rectangle whose height
is proportional to (in the chosen scale) the corresponding frequency. The
possible form of a histogram is shown in Fig. 1.9.
Calculation of empirical functions.
The data of the generated statistical
series are used to determine statistical estimates of reliability indicators,
i.e. the empirical functions:
- the distribution function of failure (estimate of FP)
ˆ
() ( /
Qt
=
nt
N
;
=
min
min
ˆ
ˆ
Qt nt N nt t N P
( )
=
=
( )/
(
, )/
;
1
1
min
1
1
ˆ
ˆ
ˆ
Qt nt N nt t
( )
=
( )/
=D
(
, )
+D
nt t N P P
( , )/
= +
;
2
2
min
1
1
2
1
2
...
k
ˆ
ˆ
∑
Qt
(
)
=
nt
(
) /
N
P
=
;
=
max
max
i
1
- the reliability function (estimate of c.d.f.) (Fig. 1.10)
ˆ
( )1
ˆ
Pt
−
=
Qt
=
( ) ;
min
min
...
ˆ
(
ˆ
Pt
−
)
=
1 (
Qt
=
)
;
max
max
- the density of distribution of failures (estimate of p.d.f.) (Fig. 1.11)
ˆ
ˆ
ft
( )
=D
nt t N t P t
+
( ,
)/
D= D
/
;
i
i i
1
i
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