McGee has a gas oven, which is inspected by inspectors according to a renewal pr

Question

McGee has a gas oven, which is inspected by inspectors 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 maintenance of the oven has not been performed for more than one year, McGee has to pay a fine of 175 dollars. McGee's strategy is to perform maintenance each time exactly one year after the visit of the inspectors. Compute the long-term amount of fine McGee has to pay per year.

Explanation / Answer

McGee will perform maintenance of his gas-oven right after an year of the inspector's visit. Let's say the inspector visited at t=0, then he will perform maintenance at t=1. Now, if the inspector visits him after t=2 years, he will be paying fine.

Let's say inspector visits him after 'x1' years.

Recall that the 'xi' is distributed uniformly over 1 to 2.5. [xi is the difference between ith and (i+1)th visit]

Fine_i = 175, if xi>2 and 0 if xi<2

E[Fine_i] = 175*(2.5-2)/(2.5-1) + 0 = $58.33

We have to find the expected value of Summation[Fine_i]/Summation[x_i] or E[Fine_i/x_i]

Fine_i/x_i = 175/x_i, if x_i>2, and 0 if x_i<2

E[Fine_i/x_i] = Integration(175/x_i*dx_i, x_i = 2 to 2.5) = 175*ln(2.5/2) = $39.05

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