# 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? $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.6667$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.5833$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.6875$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.7500$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.6250$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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# Exam P Practice Problem 98 – flipping coins

Problem 98-A

Coin 1 is an unbiased coin, i.e. when flipping the coin, the probability of getting a head is 0.5. Coin 2 is a biased coin such that when flipping the coin, the probability of getting a head is 0.6. One of the coins is chosen at random. Then the chosen coin is tossed repeatedly until a head is obtained.

Suppose that the first head is observed in the fifth toss. Determine the probability that the chosen coin is Coin 2. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.2856$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.3060$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.3295$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.3564$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.3690$ $\text{ }$ $\text{ }$ $\text{ }$

Problem 98-B

Box 1 contains 3 red balls and 1 white ball while Box 2 contains 2 red balls and 2 white balls. The two boxes are identical in appearance. One of the boxes is chosen at random. A ball is sampled from the chosen box with replacement until a white ball is obtained.

Determine the probability that the chosen box is Box 1 if the first white ball is observed on the 6th draw. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.7530$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.7632$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.7825$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.7863$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.7915$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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# Exam P Practice Problem 97 – Variance of Claim Sizes

Problem 97-A

For a type of insurance policies, the following is the probability that the size of claim is greater than $x$. $\displaystyle P(X>x) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \biggl(1-\frac{x}{10} \biggr)^6 &\ \ \ \ \ \ 0

Calculate the variance of the claim size for this type of insurance policies. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \frac{10}{7}$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \frac{75}{49}$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \frac{95}{49}$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \frac{15}{7}$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \frac{25}{7}$ $\text{ }$ $\text{ }$ $\text{ }$

Problem 97-B

For a type of insurance policies, the following is the probability that the size of a claim is greater than $x$. $\displaystyle P(X>x) = \left\{ \begin{array}{ll} \displaystyle 1 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \biggl(\frac{250}{x+250} \biggr)^{2.25} &\ \ \ \ \ \ x>0 \\ \end{array} \right.$

Calculate the expected claim size for this type of insurance policies. $\text{ }$ $\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 200.00$ $\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 203.75$ $\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 207.67$ $\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 217.32$ $\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 232.74$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$ $\text{ }$

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