context You work for an insurance company in the P&C sector for a team that takes care of car insurance. You are told that the model used for the claim amount total of each individual during a given year for a specific group of 1000 insured persons is the following: the model for the frequency is a Poisson distribution with parameter = 0.1, and that for severity is a Weibull 1 distribution with parameters = 1/2 and = 2500. For now, the insurance contracts do not include any change in coverage (no deductible, limit and coinsurance). The company faces two problems and asks for your help. Problems : Its administration costs for processing claims from this group are too high and she wishes to reduce them. Compared to the expected costs for this year, she wants to achieve an (expected) reduction of 20% the following year. To achieve this, she wants to add a deductible for each contract. Note that the fees to process each of the claims are the same and are not expected to increase from this year to the following year. You can consider that right now we are in January 2022, that all contracts started on January 1 and ends on December 31, and that they will all be renewed on January 1 January 2023 (regardless of the changes in coverage made). Your tasks are more specifically to: (a) demonstrate that the distribution of an individual's number of payments following the addition of a deductible is a Poisson distribution with parameter , where is the probability that a individual loss exceeds the deductible (consider any for the demonstration); you are asked to proceed by directly identifying the probability function; (b) identify the level of deductible that will achieve the objective; (c) identify the expected number of individuals with no payments next year, from compare to the corresponding number for that year, and interpret these results.