Environmental Engineering Reference
In-Depth Information
Q
(
t
) = 0.5 + Φ(
x
);
[1.52]
P
(
t
) = 0.5 - Φ(
x
);
[1.53]
λ(
t
) =
f
(
x
)/
S
[0.5 - Φ(
x
)].
[1.54]
Most often, when assessing the reliability of an object it is necessary
to solve the
direct problem
- at the given parameters
T
0
and
S
of the
normally distributed operating time to failure to determine a reliability
indicator (for example, c.d.f.) for the given operating time
t
. However, in
the course of design work it is also necessary to solve the
inverse problem
- determination of operating time required by the technical task for c.d.f.
of the object.
These problems are solved using the quantiles of the normalised normal
distribution.
Quantile
is the value of the random variable corresponding to a given
probability.
Denote:
t
p
- the operating time corresponding to c.d.f.
P
;
x
p
- the value of a random variable
X
corresponding to probability
P
.
Then from the constraint equation of
x
and
t
:
x
p
= (
t
p
- T
0
)/
S.
At
x
=
x
p
;
t
=
t
p
:
t
p
=
T
0
+
x
p
S
.
t
p
,
x
p
is the non-normalised and normalised quantiles of the normal
distribution, corresponding to probability
P
.
Values of the quantiles
x
p
values are given in literature for
P
≥ 0.5.
For a given probability
P
< 0.5
x
p
= -
x
1-p
.
For example, when P = 0.3
x
0.3
= -
x
1-0.3
= -
x
0.7
The probability of random value of operating time
T
fitting in a given
operating time interval [
t
1
,
t
2
] is determined by:
{
}
12
[1.55]
PT tt
∈
(,)
=
Fx Fx
( )
−
()
=Φ
( )
x
−Φ
( ,
x
2
1
2
1
where
x
1
= (
t
1
-
T
0
) /
S
,
x
2
= (
t
2
-
T
0
) /
S
.
Note that the time to failure is always positive, and the curve of c.d.f.
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