Exam P Practice Problem 69 – More Large Claim Studies

Problem 69-A

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.05^8}{5040} \ x^7 \ e^{-0.05 \ x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 80th percentile of the claim size distribution.

In a random sample of 10 claims from this portfolio of insurance policies, what is the probability that more than two of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3020

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3222

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6242

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6778

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6980

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Problem 69-B

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.04^{11}}{10!} \ x^{10} \ e^{-0.04 x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 90th percentile of the claim size distribution.

In a random sample of 15 claims from this portfolio of insurance policies, what is the probability that two or more of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.184

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.451

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.733

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.816

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Answers

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