# Exam P Practice Problem 99 – When Random Loss is Doubled

Problem 99-A

A business owner faces a risk whose economic loss amount $X$ follows a uniform distribution over the interval $0. 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 that is greater than 0.5 is paid by the insurer in full. When a loss occurs next year, determine the expected payment made by the insurer to the business owner.

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \frac{8}{16}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \frac{9}{16}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \frac{13}{16}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \frac{15}{16}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \frac{17}{16}$

$\text{ }$

$\text{ }$

$\text{ }$

Problem 99-B

A business owner faces a risk whose economic loss amount $X$ has the following density function:

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

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 that is greater than 1 is paid by the insurer in full. When a loss occurs next year, what is the expected payment made by the insurer to the business owner?

$\text{ }$

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.6667$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.5833$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.6875$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.7500$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.6250$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

$\text{ }$

probability exam P

actuarial exam

math

Daniel Ma

mathematics

expected insurance payment

deductible

$\copyright$ 2017 – Dan Ma