Tag Archives: Exponential Distribution

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}

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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}