5.3.1 Continuous
Distributions
5.3.1.1
Normal (or Gaussian) Distribution
There are two principal
applications of the normal distribution to reliability. One application deals
with the analysis of items which exhibit failure due to wear, such as
mechanical devices. Frequently the wearout failure distribution is
sufficiently close to normal that the use of this distribution for predicting
or assessing reliability is valid.
Another application is in the
analysis of manufactured items and their ability to meet specifications. No
two parts made to the same specification are exactly alike. The variability of
parts leads to a variability in systems composed of those parts. The design
must take this part variability into account, otherwise the system may not
meet the specification requirement due to the combined effect of part
variability. Another aspect of this application is in quality control
procedures.
The basis for the use of normal
distribution in this application is the central limit theorem which states
that the sum of a large number of identically distributed random variables,
each with finite mean and variance, is normally distributed.
Thus, the variations in value of
electronic component parts, for example, due to manufacturing are considered
normally distributed.
The failure density function for
the normal distribution is


(5.22) 
where:
m = the population mean
s
= the population standard deviation,
which is the square root of the variance
For most practical
applications, probability tables for the standard normal distribution are used
(See Table 5.31). The standard normal
distribution density function is given by


(5.23) 
where:
m =
0
s^{2} = 1
One converts from the normal to standard normal distribution by
using the transformations


(5.24) 


(5.25) 


(5.26) 
where:
F(t) is the cumulative
distribution function
R(t) is the reliability function
This integral cannot be evaluated in closed form; however, using
the transformations in equations 5.24 and 5.25 along with Table 5.32, the probabilities for any
normal distribution can be determined.