The number of accidents incurred by insured drivers in a given day (24 hours, starting from 6:00am) is a Poisson process with rate function 1+½t, 4, if 0 < t < 4, if 4 < t < 12, if 12 ≤ t ≤ 16, X(t) = 12t, 2, if 16 ≤ t ≤ 24 If an accident occurs, the probability that the damage amount exceeds the de- ductible is 0.25. Suppose that the amount of each claim is uniformly distributed on [$2000, $5000] and all claims are independent. (a) Find the probability that there are no accidents that are required to claim during 5pm to 8pm. (b) Find the average total amount of the claims requested in 5 hours from 8am. (c) Let N be the number of claims with claim amount less than $3000 during 10am to 2pm. Find the distribution of N. (d) It is observed that the accidents comes from seniors with probability 2/3 during 11am to 4:00pm and with probability 1/3 during other time period. Let Xt denote the number of accidents incurred by insured seniors drivers at t hours starting from 6:00am. Explain why {Xt}0

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