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

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

# Exam P Practice Problem 86 – Finding Mean and Variance

The following is the cumulative distribution function of the random variable $X$.

$\displaystyle F(x)=\left\{\begin{matrix} \displaystyle 0&\ \ \ \ \ \ x < 0 \\{\text{ }}& \\{\displaystyle \frac{(x+2)^2}{100}}&\ \ \ \ \ \ 0 \le x <6 \\{\text{ }}& \\{\displaystyle 1}&\ \ \ \ \ \ 6 \le x <\infty \end{matrix}\right.$

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Problem 86-A

Calculate the expected value of $X$.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 2.16$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.35$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 4.32$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 6.00$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 6.67$

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Problem 86-B

Calculate the variance of $X$.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 3.240$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 3.658$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 3.957$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 4.694$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 5.556$

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

# Exam P Practice Problem 85 – Supplemental Coverages for Employees

Problem 85-A

A large employer offers three supplemental benefits in addition to the major medical benefit – dental, vision and group life insurance – to its employees. The following information is known.

• Thirty five percent of the employees choose dental and vision.
• Twelve percent of the employees choose all three supplemental benefits.
• Fifty six percent of the employees choose exactly two of the three supplemental benefits.
• Sixteen percent of the employees choose neither vision nor dental.
• Five percent of the employees choose none of the three supplemental benefits.

One employee is chosen at random. What is the probability that this employee has group life insurance benefit?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.32$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.56$

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

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.65$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.68$

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Problem 85-B

A professor tracks a group of students taking astrophysics, biostatistics and chemistry. The following information is known.

• Forty five percent of the students pass astrophysics and biostatistics.
• Nineteen percent of the students pass biostatistics and chemistry only.
• Twenty two percent of the students pass neither biostatistics nor chemistry.
• Sixty seven percent of the students pass biostatistics.
• Thirty four percent of the students pass astrophysics and biostatistics only.

One student is chosen at random. What is the probability that this student passes chemistry?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.35$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.40$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.41$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.47$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.53$

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

# 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. 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 $X$ 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 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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$\copyright \ 2014 \ \ \text{ Dan Ma}$

# Exam P Practice Problem 83 – Claim Size of Auto Insurance Policies

Problem 83-A

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable $Y=X^2$ where $X$ has a normal distribution with mean 0 and variance 1.5.

What is the expected claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1250$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 1500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 1750$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 2250$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 2500$

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Problem 83-B

An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable $Y=X^2$ where $X$ has a normal distribution with mean 0 and variance 3.

What is the standard deviation of the claim size for such an auto insurance policy?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 1732$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 3000$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 4243$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 4987$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 5732$

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

# Exam P Practice Problem 82 – Estimating the Median Weight of Bears

Problem 82-A

A wildlife biologist wished to estimate the median weight of bears in Alaska. The weights of the bear population he studied follow a continuous distribution with an unknown median $M$. He captured a sample of 15 bears. Let $Y_5$ be the weight (in pounds) of the fifth smallest bear in the sample of 15 captured bears. Let $Y_{11}$ be the weight (in pounds) of the fifth largest bear in the sample.

Calculate the probability that the median $M$ is between $Y_5$ and $Y_{11}$, i.e., $P(Y_5.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.5000$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7899$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.8218$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8815$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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Problem 82-B

The wildlife biologist in Problem 82-A also wishes to estimate $\tau_{75}$, the seventy fifth percentile of the weights of bear population he studied. Let $Y_{10}$ be the weight of the tenth smallest bear in the sample of 15 captured bears. Let $Y_{14}$ be the weight of the second largest bear in the sample of 15 bears.

Calculate the probability that $\tau_{75}$ is between $Y_{10}$ and $Y_{14}$, i.e., $P(Y_{10}<\tau_{75}.

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 0.6155$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 0.7500$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 0.7715$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 0.8383$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 0.9232$

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

# Exam P Practice Problem 81 – A Medical Survey

Problem 81-A

A medical researcher tracked a group of 10,000 patients with heart disease and diabetes. The researcher found that sixty five percent of these patients have health insurance coverage. She also found that thirty percent and twenty percent of the patients with health insurance coverage have heart disease and diabetes, respectively. Of the patients with health insurance coverage, ten percent have both heart disease and diabetes. Furthermore, the medical researcher found that forty percent and twenty five percent of the patients with no health insurance coverage have heart disease and diabetes, respectively. Twenty percent of the patients with no health insurance coverage have both heart disease and diabetes.

What is the number of patients that have neither heart disease nor diabetes?

$\displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ 1500$

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 2600$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 4175$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 5825$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 8500$

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Problem 81-B

A large high school offers three sporting activities – swimming, running and basketball. Thirty percent of the students participate in swimming. Of the students participating in swimming, fifty percent participate in running, thirty percent participate in basketball and five percent participate in both running and basketball. Of the students not participating in swimming, twenty percent participate in running, forty percent participate in basketball and ten percent participate in both running and basketball.

What is the percentage of the students that participate in exactly two of the sports?

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

$\displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ 22.5 \%$

$\displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ 28 \%$

$\displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ 35 \%$

$\displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ 57.5 \%$

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$\copyright \ 2014$