Environmental Engineering Reference
In-Depth Information
f
(
t
), in general, starts from
t
= -∞ and extends to
t
= ∞.
This is not a significant disadvantage if
T
0
>>
S
, since [1.55] shows
clearly that the probability that a random variable
T
fits in the interval
P
{
T
0
- 3
S
<
T
<
T
0
+ 3
S
} ≈ 1.0 with the accuracy up to 1%. This means
that all possible values (with an error not exceeding 1%) of the normally
distributed random variable with the ratio of the characteristics
T
0
> 3S are
located in the section
T
0
± 3
S
.
When the scatter of the values of the random variable
T
is large, the
range of possible values is limited to the left (0,∞) and a truncated normal
distribution is used.
Truncated normal distribution.
It is well known that the classic normal
distribution of operating time is used efficiently at
T
0
≥ 3
S
.
For small values of
T
0
and high
S
, there may be cases in which the
c.d.f.
f
(
t
) 'covers' by its left branch the region of negative operating time
values (Fig. 1.17).
Thus, the normal distribution is a general case of distribution of the
random variable in the range (- ∞; ∞) and can be used for reliability models
only in some cases (under certain conditions).
The truncated normal distribution
is the distribution derived from the
classic normal distribution with the limited range of possible values of
operating time to failure.
In general, the truncation can be:
- left (0;
∞
);
- bilateral (
t
1
,
t
2
).
The meaning of a truncated normal distribution (TND) was considered
for the case of restricting the random value of operating time to interval
(
t
1
,
t
2
).
The density of the TND
f t cf t
()
=
(),
= −
π
c
is a normalising factor determined from the condition that the area under
the curve
f
¯
(
t
) equals 1, i.e.
1
(
tT
−
)
where
ft
( )
exp
0
2
;
2
S
S
2
t
t
t
2
2
2
∫
∫
∫
f t dt cf t dt c f t dt
()
=
()
()
=
1.
=
t
t
t
1
1
1
Therefore
1
c
=
,
t
2
∫
f t dt
()
t
1
where
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