# Exam C Practice Problem 3 – Bayesian vs Buhlmann

Problem 3-A

A portfolio of independent risks is divided into two classes. Class 1 contains 60% of the risks in the portfolio and the remaining risks are in Class 2.

For each risk in the portfolio, the following shows the distributions of the number of claims in a calendar year. $\displaystyle \begin{bmatrix} \text{ }&\text{ }&\text{Class 1} &\text{ }&\text{Class 2} \\X=x&\text{ }&P(X=x) &\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&0.50 &\text{ }&0.20 \\ 1&\text{ }&0.25&\text{ }&0.25 \\ 2&\text{ }&0.12&\text{ }&0.30 \\ 3&\text{ }&0.08&\text{ }&0.15 \\ 4&\text{ }&0.05&\text{ }&0.10 \end{bmatrix}$

A risk is randomly selected in the portfolio and is observed for two calendar years. The observed results are: 2 claim in the first calendar year and 3 claims in the second calendar year.

Determine the Bayesian expected number of claims for the selected risk in year 3. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.12$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.24$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.45$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.51$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.70$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

Problem 3-B

Using the same information as in Problem 3-A, determine the Buhlmann credibility estimate for the selected risk in year 3. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.93$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.24$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.45$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.51$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.29$

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

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