# 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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probability exam P

actuarial exam

math

Daniel Ma

mathematics

$\copyright$ 2017 – Dan Ma

# 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 94 – Tracking High School Students

Problem 94-A

A researcher tracked a group of 900 high school students taking standardized tests in math and chemistry. Some of the students were given after-school tutoring before the tests (in both subjects) and the rest of the students received no tutoring. The following information is known about the test results:

• 510 of the students passed math test and 475 of the students passed chemistry test.
• Of the students who failed both subjects, there were 20% more students who did not receive tutoring than there were students who received tutoring.
• Of the students who failed chemistry and had tutoring, there were 99 more students who failed math than there were students who passed math.
• Of the students who failed chemistry and had no tutoring, there were 4 more students who failed math than there were students who passed math.
• There were 126 students who failed math and passed chemistry.

Determine the probability that a randomly selected student from this group had tutoring given that the student passed both subjects.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.6810$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.6828$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.6859$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.6877$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.6989$

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Problem 94-B

An insurance company tracked a group of 800 insureds for 2 years. It was found that 560 of the insureds had no claims in year 1 and 380 of the insureds had no claims in year 2. Of the insureds who had no claims in both years, there were four times as many male insureds than there were female insureds. Furthermore, there were 230 male insureds who had no claims in year 2 and there were 53 females insureds who had claims in both years. It is also known that there were 85 male insureds who had claims in year 1.

Determine the number of insureds who had no claims in year 1 but had claims in year 2.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 320$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 347$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 369$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 420$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 560$

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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 91 – Reviewing a Group of Policyholders

Problem 91-A

A life insurance actuary reviewed a group of policyholders whose policies or contracts were inforce as of last year. The actuary found that 12% of the policyholders who had only a life insurance policy did not survive to this year and that 7.5% of the policyholders who had only an annuity contract did not survive to this year. The actuary also found that 5.9% of the policyholders who had both a life insurance policy and an annuity contract did not survive to this year.

In this group of policyholders, 65% of the policyholders had a life insurance policy and 57% of the policyholders had an annuity contract. Furthermore, each policyholder in this group either had a life insurance policy or an annuity contract.

What is the percentage of the policyholders that did not survive to this year?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 7.8 \%$

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

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

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 13.4 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 25.4 \%$

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Problem 91-B

A sport coach in a university tracks a group of athletes. The coach finds that 36% of the athletes who play soccer only are first year university students and that 20% of the athletes who are involved only in track and field are first year university students. The coach also discovers that 27% of the athletes participates in both soccer and track and field are first year university students.

According to university records, 45% of the athletes in this group play soccer and 68% of the athletes in this group participate in track and field. Each of the athletes in this group either plays soccer or participates in track and field.

Out of this group of athletes, what is the percentage of the athletes that are not first year university students?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 64 \%$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 67 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 70 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 74 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 80 \%$

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

# 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 89 – Finding Median

Problem 89-A

The random variables $X$ and $Y$ have the following joint density function.

$\displaystyle f(x,y)=\frac{1}{32} \ (4-x) \ \ \ \ \ \ \ 0

Suppose that $m$ is the median of $X+Y$. Which of the following is true about $m$?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.5

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 3.5

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 3.5

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Problem 89-B

The random variable $X$ has the following density function.

$\displaystyle f(x)=\frac{3}{16000} \ (400-x^2) \ \ \ \ \ \ \ 0

Suppose that $m$ is the median of $X$. Which of the following is true about $m$?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 6

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 5.5

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 5.5

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 7

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 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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$\copyright \ 2014 \ \ \text{ Dan Ma}$