## Exam C Practice Problem 26 – A Limited Fluctuation Credibility Example

Problem 26-A

You are given the following about a large portfolio of insurance policies:

• For each insurance policy, the annual number of claims follows a binomial distribution with $m$ = 3 and $q$ = 0.3.
• The claim size follows an inverse Gamma distribution with $\alpha$ = 2.1 and $\theta$ = 3.
• The number of claims and the claim sizes are independent.
• The full credibility standard has been selected so that actual claim costs will be
within 10% of expected claim costs 90% of the time.

Using limited fluctuation credibility, determine the expected number of claims required for full credibility.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 460$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 790$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2895$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3715$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4600$

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Problem 26-B

You are given the following about a large portfolio of insurance policies:

• For each insurance policy, the annual number of claims follows a binomial distribution with $m$ = 6 and $q$ = 0.1.
• The claim size follows a Gamma distribution with $\alpha$ = 0.8 and $\theta$ = 1.
• The number of claims and the claim sizes are independent.
• The full credibility standard has been selected so that actual claim costs will be
within 10% of expected claim costs 90% of the time.

Using limited fluctuation credibility, determine the expected number of exposures required for full credibility.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 514$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 582$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 970$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5141$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5818$

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

## Exam C Practice Problem 25 – Estimation Based on Claim Experience

Problem 25-A

You are given the following information about an insured population:

• For each risk in the population, the annual number of claims follows a Poisson distribution with mean $\theta$.
• The parameter $\theta$ follows a Gamma distribution with mean 2 and variance 0.8.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

• Based on the number of claims observed in the first 9 years, the Bayesian estimate for the expected number of claims per year for this risk is 4.
• Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 3.92.

What is the number of claims observed in the last year of the observation period for the selected risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7$

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Problem 25-B

You are given the following information about an insured population:

• For each risk in the population, the annual number of claims follows a Poisson distribution with mean $\theta$.
• The parameter $\theta$ follows a Gamma distribution with mean 1.6 and variance 0.32.

A risk is randomly selected from the insured population. After observing the selected risk for 10 years, the following is known.

• Based on the number of claims observed in the first 5 years, the Bayesian estimate for the expected number of claims per year for this risk is 1.7.
• Based on all the claims observed in the entire 10-year period, the Bayesian estimate for the expected number of claims per year for this risk is 1.8.

What is the number of claims observed in the last 5 years of the observation period for the selected risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 16$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 19$

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

## Exam C Practice Problem 24 – Bayesian Credibility Example

Both Problems 24-A and 24-B use the following information.

You are given the following information:

• The claim size of an insured has an exponential distribution with mean $\displaystyle \frac{1}{\theta}$.
• The parameter $\theta$ has a Gamma distribution with mean 6 and variance 12.

A randomly selected insured has one claim of size 10.

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Problem 24-A

What is the Bayesian estimate of the expected amount of the next claim for this insured?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.2$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.1$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.5$

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Problem 24-B

What is the posterior probability that the size of the next claim for this insured will be greater than 5?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.19$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

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

## Exam C Practice Problem 21 – Working with Aggregate Claims

Problem 21-A

You are given the following:

• The annual number of claims follows a Poisson distribution with mean 800.
• The claim size follows a Gamma distribution with $\alpha$ = 5 and $\theta$ = 2.
• The number of claims and the claim sizes are independent.

Using the normal approximation for the distribution of aggregate claim costs, calculate the probability that the aggregate claim costs will exceed 8350?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.17$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.20$

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Problem 21-B

You are given the following:

• The annual number of claims follows a geometric distribution with mean 8.
• The claim size follows an exponential distribution with mean 5.
• The number of claims and the claim sizes are independent.

Using the normal approximation for the distribution of aggregate claim costs, calculate the 75th percentile of aggregate claim costs?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 64$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 66$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 68$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 70$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 75$

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

## Exam C Practice Problem 17 – Estimating Claim Frequency

Both Problems 17-A and 17-B use the following information.

An insurance portfolio consists of independent risks.

For each risk in this portfolio, the number of claims in a year has a Poisson distribution with mean $\theta$. The parameter $\theta$ follows a Gamma distribution.

A risk is randomly selected from this portfolio. Prior to obtaining any claim experience, the number of claims in a year for this risk has a distribution with mean 0.6 and variance 0.72.

After observing this risk for one year, insurance company records indicate that there are 2 claims for this risk.

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Problem 17-A

After knowing the insurance company records, what is the expected number of claims per year for this risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.83$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.25$

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

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Problem 17-B

After knowing the insurance company records, what is the variance of the number of claims per year for this risk?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{26}{36}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{32}{36}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{35}{36}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{42}{36}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{45}{36}$

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

## Exam C Practice Problem 16 – Another Poisson-Gamma Problem

Both Problems 16-A and 16-B use the following information.

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and forty percent are in Class 2.

• The annual number of claims for a single risk in Class 1 follows a Poisson distribution with mean 1.
• The claim size follows a Gamma distribution with mean 1.6 and variance 1.28.
• The number of claims and the claim sizes are independent

• The annual number of claims for a single risk in Class 2 follows a Poisson distribution with mean 2.6.
• The claim size follows a Gamma distribution with mean 2.5 and variance 3.125.
• The number of claims and the claim sizes are independent

A risk is randomly selected from this portfolio. According to the records of the insurer, there are 4 claims for this risk in the amounts 2, 3, 5 and 5 within the last 3 years.

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Problem 16-A

Determine the Buhlmann credibility estimate for total claim costs for this risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.7$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.7$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.4$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4.5$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.0$

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Problem 16-B

Determine the Buhlmann credibility estimate for the number of claims for this risk in the next year.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.33$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.35$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.46$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.48$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.64$

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

## 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?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1443$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1579$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1936$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1945$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2050$

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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?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 820$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1230$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1353$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1376$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1396$

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

## Exam C Practice Problem 10 – Examples of Claim Frequency Models

Problem 10-A

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 1.6 and variance 1.28 (80% weight) and a Gamma distribution with mean 2.5 and variance 3.125 (20% weight).

A risk is randomly selected from this portfolio and observed for 3 years and is found to have incurred 4 claims. What is the probability that this risk will incur exactly 1 claim in the upcoming year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.285$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.303$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.319$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.357$

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Problem 10-B

A portfolio consists of independent risks. For each risk, the number of claims in a year has a Poisson distribution with mean $\lambda$. The parameter $\lambda$ is a mixture of a Gamma distribution with mean 2.4 and variance $\displaystyle \frac{48}{25}$ (60% weight) and a Gamma distribution with mean 3.75 and variance $\displaystyle \frac{75}{16}$ (40% weight).

A risk is randomly selected from this portfolio and observed for 2 years and is found to have incurred 3 claims.

If this risk incurs exactly 2 claims in the upcoming year, what is the probability that the given risk is from Class 2?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.205$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.214$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.263$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.275$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.300$

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Revised: May 1, 2016.
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$\copyright \ 2013-2016 \ \ \text{Dan Ma}$

## Exam C Practice Problem 9 – Examples of Claim Frequency Models

Problem 9-A

A portfolio consists of independent risks divided into two classes. Eighty percent of the risks are in Class 1 and twenty percent are in Class 2.

• For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean $\theta$ such that $\theta$ follows a Gamma distribution with mean 1.6 and variance 1.28.
• For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean $\delta$ such that $\delta$ follows a Gamma distribution with mean 2.5 and variance 3.125.

An actuary is hired to examine the claim experience of the risks in this portfolio. What proportion of the risks can be expected to incur exactly 1 claim in one year?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.26$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.27$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.28$

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Problem 9-B

A portfolio consists of independent risks divided into two classes. Sixty percent of the risks are in Class 1 and fourty percent are in Class 2.

• For each risk in Class 1, the number of claims in a year has a Poisson distribution with mean $\theta$ such that $\theta$ follows a Gamma distribution with mean 2.4 and variance $\displaystyle \frac{48}{25}$.
• For each risk in Class 2, the number of claims in a year has a Poisson distribution with mean $\delta$ such that $\delta$ follows a Gamma distribution with mean 3.75 and variance $\displaystyle \frac{75}{16}$.

An actuary is hired to examine the claim experience of the risks in this portfolio. Of the risks that incur exactly 2 claims in a year, what proportion of the risks can be expected to come from Class 2?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.38$

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

## 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?

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

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

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

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

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

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