(a) A company is analyzing the number of accidents that occur each year on the factory floor. It believes that the number of accidents per year N has a geometric distribution with parameter 0.8, so that Pr(N=n)-0.8x0.2", n=0,1,2,... For each accident, the number of employees injured is Y. where Y-X+1, and X is believed to have a Poisson distribution with parameter 2.2. The company has taken out an insurance policy, which provides a benefit of 1,000 to each injured employee, up to a maximum of three employees per accident, irrespective of the level of injury. There is no limit on the number of accidents that may be claimed for in a year. (i) Show that E(S)=0.634 and Var(S) = 2.125, where S is the total number of employees claiming benefit in a year under this policy. (8 marks) (ii) Hence find the mean and variance of the aggregate amount paid out under this policy in a year. (2 marks) (b) An insurance company has a homogeneous class of policies on which aggregate claims have a gamma distribution with parameters a=2, B=0.01. A safety loading of 25% is included in the premiums (1) (ii) Determine the adjustment coefficient, and (7 marks) The initial reserve necessary for the probability of ruin to be less than 0.02. (3 marks)