The so called bonus-malus premium calculation principle is widely spread in car insurance practice. There is a finite number of classes (tariff groups), and the premium a policy holder pays depends on what class he/she belongs to. Each year, a policy holder's class improves depending upon there were no claims during past year (a bonus: transfer to a lower class) or it deteriorates in the event of claims (a malus: transfer to a higher class class). The table below shows the premium scale (as a percentage of the premium paid by class 13 drivers) for the German bonus-malus system: 3 13 Class Premium % ClassAfter One Year (# claims) 0 1 2 3 18 200 13 18 18 18 18 17 200 13 18 18 18 18 16 175 17 18 18 18 15 175 13 16 17 18 18 14 125 13 15 17 18 18 13 100 12 14 16 17 18 12 85 11 13 14 16 18 11 70 10 13 14 16 18 10 65 9 12 13 14 18 9 60 8 11 13 14 18 8 55 7 11 13 14 18 7 50 6 11 13 14 18 6 45 5 11 13 14 18 5 40 4 10 12 13 18 4 40 3 9 11 13 18 3 40 2 8 11 13 18 2 40 1 7 11 13 18 1 40 1 7 11 13 18 Assuming that for any policy holder the collection of random variables "number of claims per year" is i.i.d. with Poisson distribution and parameter 1 = 1.6, construct the transition probability matrix for the Markov chain associated with a policy holder's class. Classify the states of the chain. If the initial condition for a certain individual is that of being in Class 11, find the probability distribution of class to which he/she belongs after 10 years. Assuming the premium paid by class 13 drivers is $400, find the the long-run average premium charged to German drivers. The so called bonus-malus premium calculation principle is widely spread in car insurance practice. There is a finite number of classes (tariff groups), and the premium a policy holder pays depends on what class he/she belongs to. Each year, a policy holder's class improves depending upon there were no claims during past year (a bonus: transfer to a lower class) or it deteriorates in the event of claims (a malus: transfer to a higher class class). The table below shows the premium scale (as a percentage of the premium paid by class 13 drivers) for the German bonus-malus system: 3 13 Class Premium % ClassAfter One Year (# claims) 0 1 2 3 18 200 13 18 18 18 18 17 200 13 18 18 18 18 16 175 17 18 18 18 15 175 13 16 17 18 18 14 125 13 15 17 18 18 13 100 12 14 16 17 18 12 85 11 13 14 16 18 11 70 10 13 14 16 18 10 65 9 12 13 14 18 9 60 8 11 13 14 18 8 55 7 11 13 14 18 7 50 6 11 13 14 18 6 45 5 11 13 14 18 5 40 4 10 12 13 18 4 40 3 9 11 13 18 3 40 2 8 11 13 18 2 40 1 7 11 13 18 1 40 1 7 11 13 18 Assuming that for any policy holder the collection of random variables "number of claims per year" is i.i.d. with Poisson distribution and parameter 1 = 1.6, construct the transition probability matrix for the Markov chain associated with a policy holder's class. Classify the states of the chain. If the initial condition for a certain individual is that of being in Class 11, find the probability distribution of class to which he/she belongs after 10 years. Assuming the premium paid by class 13 drivers is $400, find the the long-run average premium charged to German drivers