## Exam C Practice Problem 14 – Examples of Limited Fluctuation Credibility

Problem 14-A

You are given the following:

• The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
• The claim size follows a uniform distribution on $(0,t)$ where $t$ is unknown.
• The number of claims and the claim sizes are independent.

Using limited fluctuation credibility, how many expected claims are required to be 95% certain that actual claim costs will be within 5% of the expected claim costs? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1443$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1579$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1936$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1945$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2050$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 14-B

You are given the following:

• The annual number of claims generated from a portfolio of insurance policies follows a Poisson distribution.
• The claim size follows a distribution with the following moment generating function.
• $\displaystyle M(t)=\frac{1}{(1-10t)^4}$
• The number of claims and the claim sizes are independent.

What is the least number of expected claims that are required to be 90% certain that actual claim costs will be within 5% of the expected claim costs? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 820$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1230$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1353$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1376$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1396$

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

## Exam C Practice Problem 6 – Working with Posterior Distributions

Problem 6-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 the Gamma distribution with mean 2 and variance 1.

After observing this risk for five calendar years, a total of 12 claims are observed.

Which of the following is the moment generating function of the posterior distribution of $\theta$? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{2}{2-t}\biggr)^{16} \ \ \ \ \ \ \ \ t<2$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{15} \ \ \ \ \ \ \ \ t<7$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{14}{14-t}\biggr)^{9} \ \ \ \ \ \ \ \ t<14$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{15} \ \ \ \ \ \ \ \ t<2$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ M(t)=\biggl(\frac{7}{7-t}\biggr)^{16} \ \ \ \ \ \ \ \ t<7$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

Problem 6-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 Gamma distribution with mean 4 and variance $\frac{1}{2}$.

After observing this risk for eight calendar years, a total of 32 claims are observed.

Determine the coefficient of variation of the posterior distribution of $\theta$. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{64}$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{32}$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{16}$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{8}$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{4}$

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