A certain type of insurance policy has a claim rate of per year and the cover ceases and the policy expires after the first claim. Accordingly the duration of a policy is modelled by an exponential distribution with density function f x e ;0 x . A company has data on (m + n) policies which have expired and which may be assumed to be independent. Of these, m policies had duration less than 5 years and n policies had duration greater than or equal to 5 years.

(a) An investigator makes note of the actual durations x ,x ,...,x of the latter group of n

policies, but ignores the former group without even noting the value of m. In this case duration of the policy will be modelled as f x k e ;5 x . Determine the value of k, hence find the maximum likelihood value of . (b) A second investigator ignores the actual policy duration and simply notes the value of m

and n. Obtain the maximum likelihood estimate of in this case. (c) The two investigators decide to pool their data, and so have the information that there

are m policies with duration less than 5 years and n policies with actual durationsx ,x ,...,x . Obtain the maximum likelihood estimate of in this case. You may leave the results as mathematical expression. (d) You are given that m=120, n=10 and x 71 . Compute maximum likelihood value

of for case (a) and (b). You are given 0.51 for case (c). Compare your estimates under three cases.