# Exam P Practice Problem 80 – Total Insurance Payment

Problem 80-A

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the amount of loss is at least 1. Once a random loss occurs, the insurance payment to the insured is modeled by the random variable $X$ with the following density function

$\displaystyle f(x)=\frac{1}{x^2} \ \ \ \ \ 1

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.25. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.750$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.875$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.925$

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Problem 80-B

An individual purchases an insurance policy to cover a random loss. If a random loss occurs during the year, the loss amount is at least 1. Once a loss occurs, the insurance payment to the insured is modeled by the random variable $X$ with the following density function

$\displaystyle f(x)=\frac{1}{30} \ x(1+3x) \ \ \ \ \ 1

If there is a loss, there is only one loss in each year. In each year, the probability of a loss is 0.15. What is the probability that the annual amount paid to the policyholder under this policy is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.1500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.2833$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8500$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8735$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.8925$

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