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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