Page 243 - Design of Simple and Robust Process Plants

6.2 Basic Theory of Reliability 229
The standardized statistical method to define the spread is through the variance,
Var(x). For discrete distributions the variance is:
n
P 2
x i l†
Var(x) = iˆ1
n
For continuous distributions, the variance is defined by:
‡1
2 R 2
r ˆ (x±l) f(x) dx
1
where sigma (r) is the standard deviation.
There are different distributions that might describe the failure of components in
time or any other usage function, though which distribution fits the best is a matter
of evaluation. The different mathematical distributions are outlined in the following
section.

6.2.2.1 The binomial distribution
This is commonly used to quantify the probability of failures for redundant installed
systems. It is used to calculate the probability of failure on demand particular for
safeguarding systems.
For k out of n systems, the system will fail if k or more components fail. The total
failure probability is described by the cumulative binomial distribution.

P n n!
s
P (minimal k occurrences out of n trials) = p 1 p† n s
sˆk s! n s†!
where p is the probability of a system.
The mean is: l = n p, and the variance Var (x) = n p (1 ± p).
The binomial distribution has only integer numbers for occurrences.

6.2.2.2 The Poisson distribution
This distribution is used when the likelihood of an incident occurring in the near
future in not dependent on the occurrence or nonoccurrence of such an incident in
the past. This can be the case when a process is producing components some of
which do not pass the quality test (fail), and nothing is modified or changed on the
process.
The Poisson function is defined as:
x
P(x) = np† exp np†
x!
where x is the number of occurrences of a rare event (p ® 0). The Poisson distribu-
tion is a discrete probability distribution.
The mean value is: l = n p, and the variance Var(x) = n p.
The Poisson distribution gives the probability of exactly x occurrences of a rare
event with a large number of trials (n ® 1); it approximates the binomial distribu-
tion when n is large and p is small. In other words, it describes the behavior of
many rare event occurrences.