# Exam P Practice Problem 93 – Determining Average Claim Frequency

Problem 93-A

An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is $P(N=n)$ where $n=1,2,3,\cdots$. Furthermore, she finds that $P(N=n)$ is proportional to the following function: $\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=1,2,3,\cdots$

What is the weekly average number of claims arising from this group of insurance policies? $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.015$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.036$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.069$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.195$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 93-B

Let $N$ be the number of taxis arriving at an airport terminal per minute. It is observed that there are at least 2 arrivals of taxis in each minute. Based on a study performed by a traffic engineer, the probability $P(N=n)$ is proportional to the following function: $\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=2,3,4,\cdots$

What is the average number of taxis arriving at this airport terminal per minute? $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.740$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.339$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.489$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.692$

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________________________________________________________ $\copyright \ 2015 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 92 – Expected Claim Payment

Problem 92-A

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function: $\displaystyle f(x)=\frac{1}{5000} \ (100-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 12 per claim, what is the expected reduction in claim payment? $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 9.50$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 10.6$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 11.1$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 11.8$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 12.0$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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Problem 92-B

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function: $\displaystyle f(x)=\frac{1}{3200} \ (80-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 10 per claim, by what percent is the expected claim payment reduced? $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 10 \%$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 15 \%$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 22 \%$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 25 \%$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 33 \%$

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________________________________________________________ $\copyright \ 2015 \ \ \text{ Dan Ma}$