e Give 3 distributions that belong to the exponential dispersion family and can

# E give 3 distributions that belong to the exponential

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(e) Give 3 distributions that belong to the exponential dispersion family and can serve as a basis for a generalized linear model (GLM). Long answer questions: Answer these questions in the usual way, with full derivations in the booklet. [5 ] 2. The classes in a life insurance portfolio share the following characteristics: Number of claims ( N ) Claim Amount ( Y ) Expected value 10 \$5,000 Variance 10 (\$2 , 500) 2 Full credibility is granted if the average total claims amount after n years is within 10% of its expected value with 90% probability. After how many years of experience will the partial credibility factor Z of a class equal 0.54? (Round to the nearest integer.)

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ACTU 458 Final Examination December 2015 Page 2 of 3 [5 ] 3. In a particular class the observed claims in the last 4 years are X 1 = 7, X 2 = 13, X 3 = 1 and X 4 = 4. The model and prior distributions are given by f X | Θ ( x | θ ) = parenleftbigg 4 + x x parenrightbigg θ 5 (1 - θ ) x , for x = 0 , 1 , 2 , . . . , and u ( θ ) = Γ(10) Γ(6)Γ(4) θ 5 (1 - θ ) 3 , θ [0 , 1] . Derive the Bayesian premium for the 5th year. [5 ] 4. You are given that p ( x ) = x 2 in the exponential family; derive q ( θ ), μ ( θ ) and σ 2 ( θ ). Then use Jewell’s theorem to give the conjugate prior distribution, and with it calculate m , a and s 2 . [5 ] 5. Class 1 produces claims of \$100, \$1,000 or \$10,000 with probabilities 0 . 5, 0 . 3 and 0 . 2, respec- tively, while for Class 2 these probabilities are 0 . 6, 0 . 3 and 0 . 1. Class 1 has twice as many policy-holders than Class 2. A claim X 1 = 100 has been observed for a given policy-holder. Calculate the difference between the Bayesian credibility estimator of the expected value of the next claim X 2 from the same policy-holder and its B¨uhlmann’s linear credibility estimator.

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