Reliability in the period of gradual failures

246
 
 

Scattering of random variables is also convenient to characterize with 
variance 
D
= 
S
2
and the variation 
coefficient 
.
Since the integral distribution function is equal to:
then the probability of failure and the probability of failure-free operation are 
respectively equal to:
Q(t) = F(t);
P(t)
= 1 – 
F(t).
The calculation of the integrals is replaced by the use of tables for the 
so-called centered and normalized distribution, in which
m
tx
= 0 и 
S
x 
= 1.
For this distribution, the density function:
has one variable 
x
. The distribution density function is recorded in relative 
coordinates with the origin on the axis of symmetry of the loop. The distribution 
function is an integral of the distribution density:
From this equation it follows:
F
0
(x)
+ 
F
0
(–x) = 
1,
There fore
F
0
(–x)
= 1 – 
F
0
(x)
The distribution density, the probability of failure and the probability of 
failure-free operation are determined by the formulas:
Q(t)
= 
F
0
(t);
P(t)
= 1 – 
F
0
(x),
where 
quantile of the normalized normal distribution, usually 
denoted by

247
U
p
; 
f
0
(x) 
and 
F
0
(x) 
are taken from the tables.
For example:
x = 
U
p
0
1
2
3
4
 f
0
(x)
0,3989 0,2420 0,0540 0,0044 0,0001
F
0
(x)
0,5 
0,8413 09772 0,9986 0,9999
Тhe values of 
P(t)
determined by the 
formula
The time value 
t
for a given probability of failure-free operation 
P(t)
is 
determined from the dependence:
Often, instead of the integral distribution function 
F
0
(x) 
the Laplace function 
is used:
In this case:
The probability of failure and the probability of failure-free operation:
With the combined effect of sudden and gradual failures, product reliability 
for the period 
t
, if before it worked time 
T
, is:
Where 
– 
probability
of absence of sudden failures; 
– 
probability
of no gradual failures.
There are also other distributions of the random variable: a log-normal 
distribution in which the logarithm of the operating time, the gamma distri-
bution, the Weibull distribution is distributed according to the normal law, 
which is quite universal, encompassing a wide range of probabilistic variations 
by varying parameters. However, handling these distributions is more difficult.

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