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}

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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<x<100

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<x<80

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<x<4,\ 0<y<4

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

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   2.5<m<3

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   2<m<3

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   3<m<4

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   3.5<m<4

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   3.5<m<4.5

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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<x<20

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

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \   6<m<7

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \   5.5<m<6

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \   5.5<m<6.5

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \   7<m<8

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \   7<m<9

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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<x<20. An insurance policy is purchased to reimburse the loss up to a maximum limit of m where 0<m<20.

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<x<5

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<x<4

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<x<60

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<x<1. 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<x<2

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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