(a) Suppose that the loss amount follow a uniform distribution (0, 1000) and a policy with deductible amount of 200 is bought for covering that loss. Assuming that was a franchise deductible, find the mean variance of the claim amount bear by the insurance company. (18 marks)

(b) The annual number of claims on an insurance policy within a certain portfolio follows a Poisson distribution with mean . The parameter varies from policy to policy and can be considered as a random variable that follows an exponential distribution with mean . Find the unconditional distribution of the annual number of claims on a randomly chosen policy from the portfolio. (7 marks)

(c)ABC Insurance Company is having a risk of loss which is the severity is exponential distributed with mean = 500. Each policy is having an ordinary deductible of 100 per loss and a policy limit of 300 per loss. The frequency of loss event is having a negative binomial distribution with r = 2 and = 0.5. Find the variance of the insurer's annual claim payments. (18 marks)

(d)There is an ABC insurance company issue a policy with deductible of 200 and combined with policy limit of 700. The insurer collects 90% above the expected loss of this policy. The risk manager will be awarded bonus equal to 20% of the amount of premium minus annual loss claim provided if the premium charged is higher than annual loss claim. Assume annual loss claim follows an exponential distribution with mean 300. Find the expected bonus receive by the risk manager of ABC insurance company.

(e)The company named Dream Computer is selling 300,000 laptops all over the world and provide a warranty of brand new replacement for those damaged laptops which occur within one year after purchase. The average yearly cost of providing this warranty is RM600,000.00 and the standard deviation of this warranty cost is RM60,000.00. The individual claim frequency following a binomial distribution and the average cost to replace a laptop is RM3,000.00. Assume all laptops are having the same chance of being damage in the first year and all the laptop damages are independent with each other. Calculate the variance of the replacement cost for each laptop. (10 marks)