Question

3.29. The life of a car is a random variable with distribution F An individual has a policy of trading in his car either when it fails or reaches the age of A. Let R(A) denote the resale value of an A-year-old car. There is no resale value of a failed car Let C, denote the cost of a new car and suppose that an additional cost C2 is incurred whenever the car fails. (a) Say that a cycle begins each time a new car is purchased Compute the long-run average cost per unit time (b) Say that a cycle begins each time a car in use fails Compute the long-run average cost per unit time Note. In both (a) and (b) you are expected to compute the ratio of the expected cost incurred in a cycle to the expected time of a cycle The answer should, of course, be the same in both parts

Answer 1

Answer. Date:12/5/2019 (a) Clearly, Elcost per cycle] C1 F(A)R(A) +F(A)C2 and Eltime of cyclerdF(x) + A(1 - F(A) So, treating the cost as the reward, the renewal reward theorem gives im Elaccumulated cost by t Elost per cyclel C-FA)R(A)FAC2 Etime of cycle]JordF()AF(A) (b) The chance that a car fails is F(A), so the number, N, of cars bought between failures has the geometric distribution with parameter p - F(A). We have, Elcost per cycle ENC1-(N-1)R(A) C2l- C1/F(A)(1 -1/F(A)R(A)+C2

and Eltime of cycle] EI(N-1)A+Ecar lifelcar life< A F(A)A/F(A)+ adF(x)/F(A) Thus E accumulated cost bytCi/F(A)(1-1/F(A))R(A) +C2 F(A)A/F(A)+J dF()/F(A) lim Multiplying numerator and denominator by F(A) gives the same expression as in (a)