1 A simple no claims discount system for motor insurance has four levels of discount - 0%, 20%, 40% and 60%. A new policyholder starts on 0% discount. At the end of each policy year, policyholders will change levels according to the following rules: At the end of a claim-free year, a policyholder moves up one level, or remains on the maximum discount. At the end of a year in which exactly one claim was made, a policyholder drops back one level, or remains at 0%. At the end of a year in which more than one claim was made, a policyholder drops back to zero discount. For a particular driver in any year, the probability of a claim-free year is 0.9, the probability of exactly one claim is 0.075, and the probability of more than one claim is 0.025. Mike took out a policy for the first time on 1 January 2015, and by 1 January 2018 he had made only one claim, on 3 May 2017. Calculate the probability that he is on 20% discount in 2020. 2 A Markov chain is determined by the transition matrix: 0 1 0 0 0.5 0 0.5 0 Determine the period of each of the states in this chain. A Markov chain ( X,) has a discrete state space 5. The initial probability distribution is given by P[XD =/]=q;. The one-step transition probabilities are denoted by PXm+1 = im+1/Xm =im]=Pimm+1 ( m.m+1) State the Markov property for such a process. (ii) Write down expressions for the following in terms of p's and q's. (a) P[Xo = josX1 = ho..*n = in] (b ) P[X4 =1]4 A new actuarial student is planning to sit one exam each session. He expects that his performance in any exam will only be dependent on whether he passed or failed the last exam he sat. If he passes a given exam, the probability of passing the next will be o , regardless of the nature of the exam. If he fails an exam, the probability of passing the next will be B . (i) Obtain an expression for the probability that: (a) the first exam he fails is the seventh, given that he passes the first (b) he passes the fifth exam, given that he fails the first three. (fi) Explain the results above in terms of a Markov chain, specifying the state space and transition matrix. (For the purposes of this model, assume that we are only interested in predicting passing or failing, not in the number of exams passed so far.) 5 The stochastic process {X,) is defined by the relationship X, = Z + Z,_1 , where {Z; ) is a sequence of independent random variables with probability function: [1 with probability p Z1 = 1 1,000 with probability q where p+q =1 and q