Tag Archives: Frequency-Severity Model

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 22 – Estimating Aggregate Claims

Both Problems 22-A and 22-B use the following information.

You are given the following:

  • The annual number of claims for a policyholder follows a binomial distribution with mean 0.5 and variance 0.375.
  • The following is the probability function of the claim size X.
      • \displaystyle \begin{bmatrix} X=x&\text{ }&P(X=x) \\\text{ }&\text{ }&\text{ }  \\ 10&\text{ }&0.35  \\ 20&\text{ }&0.35 \\ 30&\text{ }&0.25  \\ 40&\text{ }&0.05     \end{bmatrix}

  • The number of claims and the claim sizes are independent.

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Problem 22-A

Calculate the variance of the annual aggregate claim amount for the policyholder.

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 71.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 180

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 190

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 230

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Problem 22-B

In a certain year, the policyholder has incurred at least one claim and the aggregate claim amount is below 45. Given this information, what is the mean aggregate claim amount for the policyholder?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.0

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 12.5

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 20.0

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 21.3

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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 20 – Working with Full Credibility Standard

Problem 20-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 is modeled by the random variable Y=X^2 where X has an exponential distribution with mean 2.
  • 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 5% of expected claim costs 95% of the time.

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

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3073

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4610

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6147

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7684

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9220

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Problem 20-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 is modeled by the random variable Y=4 X^2+32 where X has an exponential distribution with mean 2.
  • 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 5% of expected claim costs 95% of the time.

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

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2017

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3073

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3457

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4150

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9220

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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 following has more information about Class 1:

  • 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 following has more information about Class 2:

  • 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 15 – Examples of Frequency-Severity Models

Problem 15-A

You are given the following:

  • The annual number of claims generated for a single risk follows a Poisson distribution with mean \theta.
  • The claim size follows a uniform distribution on (0,10 \theta).
  • The risk parameter \theta is a random variable with density function \displaystyle \pi(\theta)=\frac{1}{2} \ \theta where 0<\theta<2.
  • Given \theta, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 4 claims in the amounts 5, 5, 15 and 20 within the last 5 years.

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

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.39

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.61

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9.69

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.25

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10.69

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Problem 15-B

You are given the following:

  • The annual number of claims generated for a single risk follows a Poisson distribution with mean \theta.
  • The claim size follows a uniform distribution on (0,100 \theta).
  • The prior distribution of \theta has a uniform distribution on (0,5).
  • Given \theta, the number of claims and the claim sizes are independent.

According to the records of the insurer, there are 5 claims in the total amount of 360 within the last 3 years.

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

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 97.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 164.33

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 179.33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 212.50

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 257.33

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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 13 – Working with Aggregate Claims

Both Problems 13-A and 13-B use the following information.

A portfolio if independent risks is divided into two classes. Sixty five percent of the risks are in Class 1 and thirty five percent are in Class 2.

The risks in each class are assumed to follow identical annual aggregate claim distribution. The following shows the aggregate claim distributions for the two classes.

      \displaystyle \begin{bmatrix} X=x&\text{ }&P(X=x \lvert \text{Class 1}) &\text{ }&P(X=x \lvert \text{Class 2}) \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{24}{40} &\text{ }&\displaystyle \frac{4}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 5&\text{ }&\displaystyle \frac{6}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 10&\text{ }&\displaystyle \frac{7}{40}&\text{ }&\displaystyle \frac{3}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 15&\text{ }&\displaystyle \frac{2}{40}&\text{ }&\displaystyle \frac{2}{12} \\\text{ }&\text{ }&\text{ } &\text{ }&\text{ } \\ 20&\text{ }&\displaystyle \frac{1}{40}&\text{ }&\displaystyle \frac{1}{12}    \end{bmatrix}

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Problem 13-A

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year.

What is the probability that the chosen risk will have 15 in aggregate claims in the second year?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13

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Problem 13-B

A risk is randomly selected from this portfolio and is observed to have 15 in aggregate claims in the first year and 10 in aggregate claims in the second year.

What is the probability that the chosen risk will have 15 in aggregate claims in the third year?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.09

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.10

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.11

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.13

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\copyright \ 2013 \ \ \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.

The following provides more information about these risks:

  • 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.

The following provides more information about the two classes of risks:

  • 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}