# Exam P Practice Problem 77 – Estimating Random Claim Sizes

Problem 77-A

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

$\displaystyle f(x)=\frac{3}{1000} \ (50-5x+\frac{1}{8} \ x^2), \ \ \ \ \ \ \ \ \ \ 0

What is the probability that a randomly selected claim from this insurance policy is within 120% of the mean claim size?

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.85$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.88$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.91$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.95$

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Problem 77-B

The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.

$\displaystyle f(x)=\frac{3}{2500} \ (100x-20x^2+ x^3), \ \ \ \ \ \ \ \ \ \ 0

What is the probability that a randomly selected claim from this insurance policy is within one-half of a standard deviation of the mean claim size?

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.37$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.60$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.62$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.64$

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