Environmental Engineering Reference
In-Depth Information
• probabilistic.
Statistical definitions
(
sample estimates
) are derived from the results
of reliability tests.
Suppose that the testing of a number of similar objects yields a finite
number of observations of random variable parameters of interest -
operating time to failure. These values represent a sample of a certain
volume of the general which has an unlimited amount of data on the time
to failure of the object.
Quantitative parameters defined for the general population are
true
(
probabilistic
) indicators as they objectively characterise a random variable
- time to failure.
The parameters identified for the sampling and used to draw conclusions
about a random variable are
selective
(
statistical
)
estimates
. It is obvious
that at a sufficiently large number of tests (large sample) the estimates
approach the true probabilistic parameters.
The probabilistic indicators are useful in analytical calculations, and
statistical - in experimental studies of reliability.
The statistical estimates are denoted by the ˆ sign.
The following
scheme of testing
is adopted to assess reliability.
Let
N
identical mass-produced objects are sent for testing. Test conditions
are identical, and each of these objects is tested up to failure.
The following notation is introduced:
T
= {0,
t
1
,...
t
N
} = {
t
} - a random variable of the operating time to
failure;
N
(
t
) - number of objects functioning at the operating time
t
;
n
(
t
) - number of objects that failed at the operating time
t
;
∆
n
(
t, t
+ ∆
t
) - number of objects that fail in the operatine time range
[
t, t
+ ∆
t
];
∆
t
- operating time interval.
Since the further definition of the sample estimates is based on
mathematical models of probability theory and mathematical statistics, the
following are the basic (minimum required) concepts from the theory of
probability.
Basics of mathematical models for calculations in the probability
theory
Probability theory
is a mathematical science that studies the patterns in
random phenomena.
One of the basic concepts of probability theory is the random event.
The event is any fact (the outcome) which may or may not occur as a
result of experiments (trials).
Each of these events can be associated with the number called its
probability and is a measure of the possible completion of this event.
Probability theory is based on the axiomatic approach and builds on the
concepts of set theory.
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