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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. Find the probability that no claim is made in the next one day period. [21 (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 7], 72, ..., Tan 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 7, i = 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 81 Let / be the number of claims arising on a group of policies in a period of one week and suppose that A follows a Poisson distribution with mean 60. Let X1, Xy, . . . Xy be the corresponding claim amounts and suppose that, independently of NV, these are independent and identically distributed with mean (500 and standard deviation $400. Let S = I'Mz X, be the total claim amount for the period of one week. (i) Determine the mean and the standard deviation of S. [21 (ii) Explain why the distribution of $ can be taken as approximately normal, and hence calculate, approximately, the probability that S is greater than $40,000. [3] [Total 5]A motor insurer offers a No Claims Discount scheme which operates as follows. The discount levels are {0%%. 25%. 50%. 60%). Following a claim-free year a policyholder moves up one discount level (or stays at the maximum discount). After a year with one or more claims the policyholder moves down two discount levels (or moves to, or stays in, the 0% discount level). The probability of making at least one claim in any year is 0.2. () Write down the transition matrix of the Markov chain with state space {0%, 25%, 50%, 60%} [2] (i) State, giving reasons, whether the process is: irreducible. (b) aperiodic. [2] (iin) Calculate the proportion of drivers in each discount level in the stationary distribution. The insurer introduces a "protected" No Claims Discount scheme, such that if the (0% discount is reached the driver remains at that level regardless of how many claims they subsequently make. (iv) Explain, without doing any further calculations, how the answers to parts (ii) and (iii) would change as a result of introducing the "protected" No Claims Discount scheme. (3] [Total 1 1]