## 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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1.5$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2.2$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.1$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3.5$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.15$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.19$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.21$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

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

## Exam C Practice Problem 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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3073$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4610$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6147$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7684$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9220$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2017$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3073$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3457$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4150$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 9220$

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