Ben has got a gas oven, which is inspected every now and then, according to a re

Question

Ben has got a gas oven, which is inspected every now and then, according to a renewal process with inter-arrival time distributed uniformly over the interval from one year to two years and a half. If the inspectors find out that maintainance of the oven has not been performed for more than one year, Ben has to pay a fine of 105 euros. Ben's strategy is to perform maintainance each time exactly one year after the visit of the inspectors. Compute the long-term amount of fine Ben has to pay per year.

Explanation / Answer

Taking fines for rewards, we get a renewal–reward process. Since Ben has to pay a fine with probability 1/3 each time, the expected amount of fine per inspection cycle is 35 euros. As the expected length of the inspection cycle is 7/4 years, the long-term amount of fine per year equals:

Fine = 35/(7/4)

Fine = 20 Euros

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