Exam C Practice Problem 4 – Buhlmann Credibility Examples

Problem 4-A

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has a uniform distribution on (0.5,2.5).

After observing for three calendar years, this risk is found to have incurred 1 claim in year 1, 2 claims in year 2 and 3 claims in year 3.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk in year 4.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.65

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.70

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.97

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.00

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Problem 4-B

You are given the following:

    • The number of claims in a calendar year for a given risk follows a Poisson distribution with mean \theta.
    • The prior distribution of \theta has the following density function.
      • \displaystyle \pi(\theta)=\frac{1}{2} \ (2-\theta), \ \ \ \ \ \ 0<\theta<2

The given risk is observed for 6 calendar years and is found to have incurred a total of 10 claims.

Determine the Buhlmann credibility estimate for the expected claim frequency for the given risk for the next calendar year.

\text{ }

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{62}{57}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{3}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{3}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2

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

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