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Smith works on a temporary basis. The mean length of each job he gets is three months. If the amount of time he spends between jobs is exponentially distributed with mean 2, then at what rate does Mr. Smith get new jobs? 3. Each time a machine is repaired it remains up for an exponentially distributed time with rate . It then fails, and its failure is either of two types. If it is a type 1 failure, then the time to repair the machines is exponential with rate 1 ; if it is a type 2 failure, then the repair time is exponential with rate 2 . Each failure is , independently of the time it took the machines to fail, a type 1 failure with probability p and a type 2 failure with probability 1 P. What proportion of time is the machine down due to a type 1 failure? What proportion of time is the machine down due to a type 2 failure? What proportion of time is it up?

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