Address the following questions appropriately including explanations in your working.

Claims under a certain type of insurance policy occur as a Poisson process so that the number of claims arising from a policy in one calendar year is a Poisson random variable with mean u. In order to estimate / a random sample of n such policies is examined and it is found that no of these policies incurred no claims during the last full calendar year, m of them incurred one claim and the remainder incurred more than one claim. (i) Write down the likelihood of these observations and show that the maximum likelihood estimator / is a solution of the equation: 1 - (no +n ) u+ e-" (nu2 + nou-m)=0 [ You do not need to verify that a maximum is attained.] [10] (ii) A random sample of 20 such policies yields no =8 and m =7. By using the tables of the cumulative Poisson distribution, or otherwise, to find a good starting approximation for / and then using trial and error (or a more sophisticated method), determine / correct to two decimal places. [7] [Total 17]A life table with a select period of 2 years is based on rates of mortality, which satisfy the following relationship: 1 9x-r]+r - 3-1 -Xqx (for all values of x , and r = 0,1) 960 = 0.0195 , 461 = 0.0198, 962 = 0.0200 and /63 = 100,000 (i) Calculate: (a) 62 (b) 460]+1 (c) (60] [4] (ii) A select life aged 60 subject to the mortality table described in (i) above purchases a 3-year endowment assurance with sum assured f10,000. Premiums of 3,000 are payable annually throughout the term of the policy or until earlier death. The death benefit is payable at the end of the year of death. Calculate the expected value of the present value of the profit or loss to the office on the contract, assuming an effective rate of interest of 6% pa, and ignoring expenses. [4] [Total 8]