Exam P Practice Problem 84 – When Random Loss is Doubled

Problem 84-A

A business owner faces a risk whose economic loss amount X follows a uniform distribution over the interval 0<x<1. In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable Y=2X.

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 0.5 is the responsibility of the business owner and any loss amount in excess of 0.5 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{4}{16}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{6}{16}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{7}{16}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{16}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{16}

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Problem 84-B

A business owner faces a risk whose economic loss amount X whose density function is:

    \displaystyle f(x)=\frac{x}{2} \ \ \ \ \ \ 0<x<2

In the next year, the loss amount is expected to be doubled and is expected to be modeled by the random variable Y=2X.

Suppose that the business owner purchases an insurance policy effective at the beginning of next year with the provision that any loss amount less than or equal to 1 is the responsibility of the business owner and any loss amount in excess of 1 is the responsibility of the insurer. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{5}{12}

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{2}{3}

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{19}{12}

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{27}{16}

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{21}{12}

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Answers

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\copyright \ 2014 \ \ \text{ Dan Ma}

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