Suppose that the time 7, measured in days, until the next claim arises under a portfolio of non-life insurance policies, follows an exponential distribution with mean 2. (i) Find the probability that no claim is made in the next one day period. [2] (ii) The median of a random variable is defined as the value for which the cumulative distribution function of the variable is equal to 0.5. Find the median time until the next claim arises. [2] (iii) Now let T1, T2, ..., T3 be the times (in days) until the next claim arises under cach one of 30 similar portfolios of non-life insurance policies, and assume that each Ti=1.....30, follows an exponential distribution with mean 2. independently of all others. Calculate, approximately, the probability that the total of all 30 times which elapse until a claim arises on each of the portfolios exceeds 45 days. [4] [Total 8] Let N be the number of claims arising on a group of policies in a period of one week and suppose that N follows a Poisson distribution with mean 60. Let X1, X2, X be the corresponding claim amounts and suppose that, independently of N, these are independent and identically distributed with mean 500 and standard deviation 400. N Let SX, be the total claim amount for the period of one week. (i) Determine the mean and the standard deviation of S. (ii) [2] Explain why the distribution of S can be taken as approximately normal, and hence calculate, approximately, the probability that S is greater than 40,000. [3] [Total 5]