Exam c Practice Problem 7 – Working with Buhlmann Credibility

Problem 7-A

You are given the following information:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The risk parameter \theta follows a Gamma distribution whose coefficient of variation is 0.5.
    • After observing the given risk for 5 calendar years, twelve claims are observed.
    • Based on the observed data, the posterior distribution of \theta is a continuous distribution whose mean is 2.0.

What is the Buhlmann credibility used in estimating the expected claim frequency for the given risk in the next period?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.500

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.600

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.625

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.650

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.667

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Problem 7-B

You are given the following information:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The risk parameter \theta follows a Gamma distribution with mean 1.5.
    • The value of Buhlmann’s k is 8.
    • After observing the given risk for 4 calendar years, the posterior distribution of \theta is a continuous distribution whose mean is 1.75

What is the number of claims observed in the observation period?

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 11

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\copyright \ 2013 \ \ \text{Dan Ma}

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