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 , 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  and update it by taking to account that you found no unscheduled shutdowns, under the maintenance schedule you found in (b) for 3 months.