Exercise1: A critical part used on a manufacturing machine has an exponential failure distribution withamean of 1000 (operating) days. When the part fails,it is immediately replaced with a spare. All spares must be purchased nowsince the parts supplier willterminateproduction. The life of the machine is 3,650 (operating) days.

1.Explain whyPoisson distribution can represent the number of part failures during the machines life.Give the Poisson distribution mean.

2.If the manufacturer decides to buyfoursparesfrom the supplier,Findthe probability of not running out of parts(causing the machine failure)during the machine life of 3,650 days.

3.How many spares should be purchased toguarantee a99% reliability of no stock out resulting in machine failure?Why?

Exercise 2: The failure distribution of a rotor is Weibull,withandyears. Assume that the rotor cannot be repaired but must be replaced on failure.

1.Find the MTTF of the rotor. The MTTF of Weibull life distribution is:, whereis GAMMA Excel function.

2.A 1-year warranty is available. Compute the probability of a failure occurring during the first year.

3.The first year did pass without a failure. What istheprobability of observing a failure in the second year?(Hint:you can use Bayes rule)

Exercise 3:A pump has an output flow of250gallons per minute.In the system, three pumps of such type workactively to maintain a required flow of 500gallons per minuteor more at the system outlet. The pumps have a time to failure with Weibull distribution ofandyears.

1.Find the reliability of pump at 1 year of operation.

2.Find the system (consisting of 3 pumps) reliability at 1 year of operation.