hello,, please assist me tackle these questions

A motor insurance company offers annually renewable policies. To encourage policyholders to renew each year it offers a No Claims Discount system which reduces the premiums for those people who claim less often. There are four levels of premium: C: no discount 1: 15% discount 2: 25% discount 3: 40% discount A policyholder who does not make a claim in the year, moves up une level of discount the following year (or stays at the maximum level). A policyholder who makes one or more claims in a year moves down one level of discount if they did not claim in the previous year (or remains at the lowest level) but if they made at least one claim in the previous year they move down two levels of discount (subject to not going below the lowest level). (a) Explain how many states are required to model this as a Markov chain. (b) Draw the transition graph of the process. [3] The probability, p, of making at least one claim in any year is constant and independent of whether a claim was made in the previous year. (ii) Calculate the proportion of policyholders who are at the 25% discount level in the long run given that the proportion at the 40% level is nine times that at the 15% level [6] (iii) (a) Explain how the state space of the process would change if the probability of making a claim in any one year depended upon whether a claim was made in the previous year. (b) Write down the transition matrix for this new process. 141 [Total 13]The total number of claims / on a portfolio of insurance policies has a Poisson distribution with mean .. Individual claim amounts are independent of / and each other, and follow a distribution X with mean p and variance o'. The total aggregate claims in the year is denoted by S. The random variable S therefore has a compound Poisson distribution. (i) Derive an expression for the moment generating function of $ in terms of the moment generating function of X. [4] (ii) Derive expressions for the mean and variance of S in terms of 1, p and o. [6] For a particular type of policy, individual losses are exponentially distributed with mean 100. For losses above 200 the insurer incurs an additional expense of 50 per claim. (iii) Calculate the mean and variance of $ for a portfolio of such policies with A = 500. [9] [Total 19]