5.5.1.2 __Bayes’ Example
(Continuous Distribution)__

As with the discrete example, the basic equation can be extended
to cover continuous probability distributions. For example, assume that based
upon prior test results, engineering judgment, etc. it has been observed that
r failures occur in time t. The probability density function of t is a gamma
distribution given by

where:

t is the amount of testing time (scale parameter)

r is the
number of failures (shape parameter)

From Section 5.3.5, we know that (note changes in
notation)

and

Eqs.
(5.64 and 5.65) represent the prior failure rate and the prior variance. Let
us assume that these are given by 0.02 and (0.01)^{2} , respectively.
Assume that we then run a reliability test for 500 hours (t’) and observe 14
failures (r’). What is the posterior estimate of failure
rate?

The
basic expression for the continuous posterior distribution is given
by

where:

f(l) is the prior distribution of l, Eq. (5.65)

f(t|l) is the sampling
distribution of t based upon the new data

f(l|t) is the posterior
distribution of combining the prior distribution and the new
data.

It can be shown that the
posterior distribution resulting from performing the operations indicated in
Eq. (5.68) is

which is another gamma
distribution with

shape parameter = (r +
r’)

scale parameter = (t +
t’)

Using Eqs. (5.66) and (5.67) to solve for r and t, we obtain

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Therefore,

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Returning to the posterior gamma distribution, Eq. (5.69) we
know that the posterior failure rate is

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From the test data r’ = 14, t’ = 500, and we found that r = 4,
and t = 200; thus

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This compares with the traditional estimate of failure rate from
the test result, 14/500 = 0.028. Thus, the use of prior information resulted
in a failure rate estimate lower than that given by the test
results.