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An investigation was carried out into the relationship between sickness and mortality in an historical population of working class men. The investigation used a three-state model with the states: Healthy Sick Dead Let the probability that a person in state i at time I will be in state j at time atf be Pr . Let the transition intensity at time x+r between any two states i and ) be pxe - Draw a diagram showing the three states and the possible transitions between them. [2] (ii) Show from first principles that or 23 =, Px Uxtrex Hitt- [5] (iii) Write down the likelihood of the data in the investigation in terms of the transition rates and the waiting times in the Healthy and Sick states, under the assumption that the transition rates are constant. The investigation collected the following data: man-years in Healthy state 265 man-years in Sick state 140 number of transitions from Healthy to Sick 20 number of transitions from Sick to Dead 40 (iv) Derive the maximum likelihood estimator of the transition rate from Sick to Dead. [3] (v ) Hence estimate: (a) the value of the constant transition rate from Sick to Dead (b ) 95 per cent confidence intervals around this transition rate 141 [Total 17]A motor insurance company operates a No Claims Discount scheme with discount levels 0%, 25% and 50% of the annual premium of 1,000. The probability of having an accident during any year is 0.I (ignore the possibility of more than one accident in a year). The policyholder moves one level up in the discount scheme (or stays at the 50% level) in the event of a claim free year and moves one level down (or stays at the 0 level) if the claim does not involve a criminal offence. If the claim involves a criminal offence then the policyholder automatically moves to the 0% discount level. One in ten accidents involves a criminal offence. The policyholder makes a claim only if the cost of repairs is higher than the aggregate additional premiums payable in the next two claim-free years Calenlate, for each level of discount, the cost of a repair below which the policyholder will not claim, distinguishing between claims that involve a criminal offence and claims that do not. [4] (ii) Calculate the probability of a claim for each level of discount given that an accident has occurred, given that the repair cost following an accident has an exponential distribution with mean 400. [5] (iii) Calculate the stationary distribution of the proportion of policyholders at cach discount level. (7] [Total 16]