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# Poisson Process

A certain piece of machinery is known to fail according to a Poisson process.

a) In a series of tests, the piece was let operate till failure, repaired immediately, and let operate till next failure and so on for 3 months. The total number failures observed were 3. If the intent of the test was to determine &#61548;, was the test run for too long, too short or just about the appropriate length of time ?

b) To minimize unscheduled shutdowns, the piece of machinery is to be inspected and maintained on a regular interval of X days. What X should you select to have 90% confidence that you will not see failures during operation?

c) Based on the results from (b), estimate the likelihood of finding no failure in 2 consecutive time periods.

d) Based on your results from (a) assign a probability distribution to &#61548; and update it by taking to account that you found no unscheduled shutdowns, under the maintenance schedule you found in (b) for 3 months.

#### Solution Preview

A certain piece of machinery is known to fail according to a Poisson process.

a) In a series of tests, the piece was let operate till failure, repaired immediately, and let operate till next failure and so on for 3 months. The total number failures observed were 3. If the intent of the test was to determine &#61548;, was the test run for too long, too short or just about the appropriate length of time ?

To answer this question, we need to evaluate the confidence interval around the observed failure rate.

The observed failure rate is simply &#61548;e = re/T = 3/3 = 1 month-1.

Using the method introduced in the handout "More on confidence interval" for Poisson process, we find the 90% confidence interval for &#61548;e is from 0.273 to 2.58. The lower limit is ~ 3 times smaller than the observed value, while the upper limit is ~3 times higher ...

#### Solution Summary

A certain piece of machinery is known to fail according to a Poisson process.

a) In a series of tests, the piece was let operate till failure, repaired immediately, and let operate till next failure and so on for 3 months. The total number failures observed were 3. If the intent of the test was to determine &#61548;, was the test run for too long, too short or just about the appropriate length of time ?

b) To minimize unscheduled shutdowns, the piece of machinery is to be inspected and maintained on a regular interval of X days. What X should you select to have 90% confidence that you will not see failures during operation?

c) Based on the results from (b), estimate the likelihood of finding no failure in 2 consecutive time periods.

d) Based on your results from (a) assign a probability distribution to &#61548; and update it by taking to account that you found no unscheduled shutdowns, under the maintenance schedule you found in (b) for 3 months.

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