# Exam P Practice Problem 102 – estimating claim costs

Problem 102-A

Insurance claims modeled by a distribution with the following cumulative distribution function.

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 0 < x \le 4 \\ \text{ } & \text{ } \\ \displaystyle 1-\frac{2}{3} x+\frac{1}{8} x^2- \frac{1}{1536} \ x^4 &\ \ \ \ \ \ 4 < x \le 8 \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ x > 8 \\ \end{array} \right.$

The insurance company is performing a study on all claims that exceed 3. Determine the mean of all claims being studied.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.8$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 4.9$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 5.0$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5.1$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 5.2$

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Problem 102-B

Insurance claims modeled by a distribution with the following cumulative distribution function.

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x \le 0 \\ \text{ } & \text{ } \\ \displaystyle \frac{1}{50} \ x^2 &\ \ \ \ \ \ 0 < x \le 5 \\ \text{ } & \text{ } \\ \displaystyle -\frac{1}{50} x^2+\frac{2}{5} x- 1 &\ \ \ \ \ \ 5 < x \le 10 \\ \text{ } & \text{ } \\ \displaystyle 1 &\ \ \ \ \ \ x > 10 \\ \end{array} \right.$

The insurance company is performing a study on all claims that exceed 4. Determine the mean of all claims being studied.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 5.9$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 6.0$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 6.1$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 6.2$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.3$

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# Exam P Practice Problem 100 – find the variance of loss in profit

Problem 100-A

The monthly amount of time $X$ (in hours) during which a manufacturing plant is inoperative due to equipment failures or power outage follows approximately a distribution with the following moment generating function.

$\displaystyle M(t)=\biggl( \frac{1}{1-7.5 \ t} \biggr)^2$

The amount of loss in profit due to the plant being inoperative is given by $Y=12 X + 1.25 X^2$.

Determine the variance of the loss in profit.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,927.20}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \text{279,608.20}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \text{475,693.76}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \text{583,358.20}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \text{601,769.56}$

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Problem 100-B

The weekly amount of time $X$ (in hours) that a manufacturing plant is down (due to maintenance or repairs) has an exponential distribution with mean 8.5 hours.

The cost of the downtime, due to lost production and maintenance and repair costs, is modeled by $Y=15+5 X+1.2 X^2$.

Determine the variance of the cost of the downtime.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \text{130,928.05}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \text{149,368.45}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \text{181,622.05}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \text{188,637.67}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \text{195,369.15}$

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# 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.

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

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

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

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

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

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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?

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

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

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

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

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

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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.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \frac{10}{7}$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \frac{75}{49}$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \frac{95}{49}$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \frac{15}{7}$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \frac{25}{7}$

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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.

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$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 200.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 203.75$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 207.67$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 217.32$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 232.74$

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# Exam P Practice Problem 96 – Expected Insurance Payment

Problem 96-A

An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount $X$ is:

$\displaystyle F(x) = \left\{ \begin{array}{ll} \displaystyle 0 &\ \ \ \ \ \ x<0 \\ \text{ } & \text{ } \\ \displaystyle \frac{3}{25} \ x^2 - \frac{2}{125} \ x^3 &\ \ \ \ \ \ 0 \le x<5 \\ \text{ } & \text{ } \\ 1 &\ \ \ \ \ \ 5

Given a random loss $X$, determine the expected payment made under this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 1.54$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 1.72$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 4.63$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 6.26$

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Problem 96-B

An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount $X$ is:

$\displaystyle f(x) = \left\{ \begin{array}{ll} \displaystyle \frac{3}{8} \biggl(1- \frac{1}{4} \ x + \frac{1}{64} \ x^2 \biggr) &\ \ \ \ \ \ 0

Given a random loss $X$, what is the expected benefit paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.51$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.57$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.63$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 1.60$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 2.00$

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$\copyright \ 2016 - \text{Dan Ma}$

# Exam P Practice Problem 93 – Determining Average Claim Frequency

Problem 93-A

An actuary performs a claim frequency study on a group of auto insurance policies. She finds that the probability function of the number of claims per week arising from this set of policies is $P(N=n)$ where $n=1,2,3,\cdots$. Furthermore, she finds that $P(N=n)$ is proportional to the following function:

$\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=1,2,3,\cdots$

What is the weekly average number of claims arising from this group of insurance policies?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.015$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.036$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.069$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.195$

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Problem 93-B

Let $N$ be the number of taxis arriving at an airport terminal per minute. It is observed that there are at least 2 arrivals of taxis in each minute. Based on a study performed by a traffic engineer, the probability $P(N=n)$ is proportional to the following function:

$\displaystyle \frac{e^{-2.9} \cdot 2.9^n}{n!} \ \ \ \ \ \ \ n=2,3,4,\cdots$

What is the average number of taxis arriving at this airport terminal per minute?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.740$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2.900$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.339$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.489$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.692$

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$\copyright \ 2015 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 92 – Expected Claim Payment

Problem 92-A

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

$\displaystyle f(x)=\frac{1}{5000} \ (100-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 12 per claim, what is the expected reduction in claim payment?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 9.50$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 10.6$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 11.1$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 11.8$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 12.0$

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Problem 92-B

The size of a claim that an auto insurance company pays out is modeled by a random variable with the following density function:

$\displaystyle f(x)=\frac{1}{3200} \ (80-x) \ \ \ \ \ \ \ 0

By subjecting the insured to a deductible of 10 per claim, by what percent is the expected claim payment reduced?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 10 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 15 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 22 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 25 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 33 \%$

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$\copyright \ 2015 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 90 – Insurance Benefits

Problem 90-A

A random loss follows an exponential distribution with mean 20. An insurance reimburses this random loss up to a benefit limit of 30.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 4.5$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 5.1$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 6.3$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 8.5$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 11.2$

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Problem 90-B

A random loss follows an exponential distribution with mean 100. An insurance reimburses this random loss up to a benefit limit of 200.

When a loss occurs, what is the expected value of the benefit not paid by this insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 12.6$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 13.5$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 24.6$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 40.6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 40.7$

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

# Exam P Practice Problem 88 – Expected Value of Insurance Payments

Problem 88-A

A random loss $X$ has a uniform distribution over the interval $0. An insurance policy is purchased to reimburse the loss up to a maximum limit of $m$ where $0.

The expected value of the benefit payment under this policy is 8.4. Calculate the value of $m$.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 8.7$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 9.0$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 12.0$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 13.6$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 18.3$

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Problem 88-B

An individual purchases an insurance policy to cover a loss $X$ whose density function is:

$\displaystyle f(x)=\frac{2}{25} \ (5-x) \ \ \ \ \ \ \ \ 0

The insurance policy reimburses the policy owner up to a benefit limit of 4 for each loss. What is the expected value of insurance payment made to the policy owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1.35$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 1.41$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 1.49$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 1.65$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 1.67$

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

# Exam P Practice Problem 87 – Modeling Insurance Payments

Problem 87-A

A business owner is facing a risk whose economic loss is modeled by the random variable $X$. The following is the density function of $X$.

$\displaystyle f(x)=\frac{1}{8} \ (4-x) \ \ \ \ \ \ \ \ 0

The business owner purchases an insurance policy to cover this potential loss. The insurance policy pays the business owner 80% of the amount of each loss.

Given that a loss has occurred, what is the probability that the amount of the insurance payment to the business owner is less than 2?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.25$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.36$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.64$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.75$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.86$

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Problem 87-B

An individual purchases an insurance policy to cover a loss $X$ whose density function is:

$\displaystyle f(x)=\frac{1}{1800} \ x \ \ \ \ \ \ \ \ 0

The insurance policy reimburses the policy owner 50% of each loss. Given that a loss has occurred, what is the median amount of the insurance payment made to the policy owner?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 15.00$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 18.65$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 21.21$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 23.63$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 42.43$

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