Environmental Engineering Reference
In-Depth Information
n
=
[1.16]
Expression [1.16] is called the formula for the probabilities of future
events.
PA A
( |
)
PH APA HA
( |
)( |
.
1
i
1
i
i
=
1
1.2.2 The cumulative distribution function, probability
density function, failure rate 
Statistical evaluation of the cumulative distribution function (c.d.f.) -
empirical reliability function - is defined by the ratio of the number N ( t )
of objects which worked flawlessly up to operating time t , to the number of
objects repaired up to the beginning of the tests ( t = 0) and the total number
of objects N :
Nt
()
ˆ ()
Pt
=
.
[1.17]
N
Assessment of c.d.f. can be regarded as an indicator of the proportion of
good state terms at the operating time t .
Since N ( t ) = N - n ( t ), then c.d.f. from [1.17] is
nt
()
ˆ () 1
ˆ
Pt
=−=−
1
Qt
(),
[1.18]
N
where Q ( t ) = n ( t ) / N is the estimate of failure probability (FP).
In statistical evaluation the FP estimate is the empirical distribution
function of failures.
Since the events consisting in the occurrence or non-occurrence of failure
at operating time t , are opposite, then
￿ ￿ ￿ ￿ ￿
ˆ ()
ˆ
[1.19]
Pt Q =
() 1.
=
It is easy to verify that c.d.f. is decreasing and FP increasing function
of operating time. In fact:
• at the beginning of trial t = 0, the number of working objects is
equal to their total number N ( t ) = N (0) = N , and the number of
failed objects is n ( t ) = n (0) = 0, so
ˆ
ˆ
ˆ
ˆ
Pt P
( )
=
(0)
1 and
=
Qt Q
( )
(0)
0;
=
=
• at the service life t →∞ all the objects put to the test fail, i.e.
N (∞) = 0 and n (∞) = N , so
ˆ
ˆ
ˆ
ˆ
Pt P
( )
= ∞=
(
)
0 and
Qt Q
( )
= ∞=
(
)
1.
• Probabilistic determination of c.d.f.
P ( t ) = P { T t }.
[1.20]
Thus, the c.d.f. is the probability that a random value of the operating
time to failure T will not be less than some specified operating time t .
 
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