Tag Archives: Poisson distribution

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 67 – Statistical Studies of Insured Drivers

Problem 67-A

An auto insurance company performed a statistical study on its insured drivers. The following table shows the results.

            \displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim}  \\\text{ }&\text{ }&\text{ } \\ \text{16-20}&\text{ }&15 \% &\text{ }&0.18  \\\text{ }&\text{ }&\text{ } \\ \text{21-30}&\text{ }&20 \% &\text{ }&0.12 \\\text{ }&\text{ }&\text{ } \\ \text{31-50}&\text{ } &30 \% &\text{ }&0.08 \\\text{ }&\text{ }&\text{ } \\ \text{51-70}&\text{ }&25 \% &\text{ }&0.09 \\\text{ }&\text{ }&\text{ } \\ \text{71 and up}&\text{ }&10 \% &\text{ }&0.11  \end{bmatrix}

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution.

Suppose that an insured driver in the study had exactly 2 claims in the past year. What is the probability that the insured driver is from the age group 16-20?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.150

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.223

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.249

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.376

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.415

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Problem 67-B

An auto insurance company performed a statistical study on its younger insured drivers (under 35 years of age). The following table shows the results.

            \displaystyle \begin{bmatrix} \text{Age Group}&\text{ }&\text{Percentage}&\text{ }&\text{Annual Probability of} \\ \text{ }&\text{ }&\text{of its Drivers}&\text{ }&\text{At Least One Claim}  \\\text{ }&\text{ }&\text{ } \\ \text{16-17}&\text{ }&12 \% &\text{ }&0.18  \\\text{ }&\text{ }&\text{ } \\ \text{18-24}&\text{ }&38 \% &\text{ }&0.10 \\\text{ }&\text{ }&\text{ } \\ \text{25-34}&\text{ } &50 \% &\text{ }&0.06  \end{bmatrix}

The authors of the statistical study also found that for any insured driver in the study, the annual number of claims follows a Poisson distribution. Furthermore, for any insured driver in the study, the number of claims in one year is independent of the number of claims in any other year.

Suppose that in a 2-year period, an insured driver in the study had exactly 1 claim in year 1 and exactly 2 claims in year 2. What is the probability that the insured driver is from the age group 16-17?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.120

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.229

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.241

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.329

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.576

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Exam P Practice Problem 65 – Total Insurance Payment

Problem 65-A

The number of random losses in a calendar year for an individual has a Poisson distribution with mean 1. When a loss occurs, the individual loss amount is either 2 or 4, with probabilities 0.6 and 0.4, respectively.

When multiple losses occur for this individual, the individual loss amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 1 per loss.

In the next calendar year, let S be the total payment made by the insurance company to the insured. Calculate P(2 \le S \le 4).

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.12204

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.16974

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29380

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31689

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34277

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Problem 65-B

The number of claims in a calendar year for an insured has a Poisson distribution with mean 1.2. When a claim occurs, the individual claim amount is either 10 or 20, with probabilities 0.8 and 0.2, respectively.

When multiple claims occur for this insured, the individual claim amounts are independent.

This individual purchases an insurance policy to cover the random losses with a deductible of 5 per loss.

In the next calendar year, let S be the total payment made by the insurance company to the insured. Calculate P(10 \le S < 30).

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1010

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2986

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3709

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3826

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3906

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Exam P Practice Problem 63 – Total Minutes of Telephone Calls

Problem 63-A

For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 12. The length in time (in minutes) of each telephone call has an exponential distribution with mean 5 minutes.

The length of time of one telephone call is independent of the length of time of any other telephone call.

On a given day, this individual makes or receives 4 telephone calls. What is the probability that this person is on the telephone for more than half an hour?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1218

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1260

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1456

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1490

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1512

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Problem 63-B

For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 16. The length in time (in minutes) of each telephone call has an exponential distribution with mean 8 minutes.

The length of time of one telephone call is independent of the length of time of any other telephone call.

On a given day, this individual makes or receives 5 telephone calls. What is the probability that this person is on the telephone for more than 45 minutes?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2237

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2596

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3384

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3975

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4085

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Exam P Practice Problem 53 – Hospital Expense Plans

Problem 53-A

An insurer sells a hospital expense plan that pays a fixed sum per day of hospitalization. Suppose that the number of days of hospitalization in a year for someone insured under this plan has a Poisson distribution with mean 0.8.

In each calendar year, the plan pays 2,000 for each day of hospitalization subject to the condition that the first two days of hospitalization are the responsibilities of the insured.

What is the expected payment for hospitalization during a calendar year under this hospital expense plan?

      \displaystyle (A) \ \ \ \ \ \ \ 116.24

      \displaystyle (B) \ \ \ \ \ \ \ 244.75

      \displaystyle (C) \ \ \ \ \ \ \ 305.93

      \displaystyle (D) \ \ \ \ \ \ \ 785.26

      \displaystyle (E) \ \ \ \ \ \ \ 1600.00

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Problem 53-B

An insurer sells a hospital expense plan that pays a fixed sum per day of hospitalization. Suppose that the number of days of hospitalization in a year for someone insured under this plan has the following probability function.

      \displaystyle P(X=x)=\frac{3}{4^{x+1}} \ \ \ \ \ \ \ \ x=0,1,2,3,\cdots

In each calendar year, the plan pays 1,000 for each day of hospitalization subject to the condition that the first day of hospitalization is the responsibility of the insured.

What is the expected payment for hospitalization during a calendar year under this hospital expense plan?

      \displaystyle (A) \ \ \ \ \ \ \ 76.83

      \displaystyle (B) \ \ \ \ \ \ \ 83.33

      \displaystyle (C) \ \ \ \ \ \ \ 111.11

      \displaystyle (D) \ \ \ \ \ \ \ 145.83

      \displaystyle (E) \ \ \ \ \ \ \ 333.33

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Exam P Practice Problem 44 – Traffic Statistics Using Poisson Distribution

Problem 44-A

The number of car accidents in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 4 per week. The number of car accidents in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8 per week.

Assume that on a weekly basis, the number of accidents in one highway is independent of the number of accidents in the other highway.

In one particular week, exactly 5 auto accidents took place in these two highways. What is the probability that Highway #1 had exactly 2 accidents in this particular week?

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Problem 44-B

The number of cars (in a day) that break down in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 16. The number of cars (in a day) that break down in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8.

Assume that on a daily basis, the number of cars breaking down in one highway is independent of the number of cars breaking down in the other highway.

In one particular day, exactly 8 cars were found to break down in these two highways. What is the probability that Highway #1 had exactly 5 cars breaking down in this particular day?

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Exam P Practice Problem 38 – Poisson Distribution

Problem 38-A

Two customers (Customer #1 and Customer #2) just purchased identical insurance coverage. The number of claims for each insured is assumed to follow a Poisson distribution with mean 1.5 per year. Assume that the number of claims for Customer #1 is independent of the number of claims for Customer #2.

What is the probability that in the coming year, Customer #1 will have exactly one claim and Customer #2 will have exactly two claims?

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Problem 38-B

The number of customers visiting a jewelry store on a weekday has a Poisson distribution with mean 4 per hour. Assume that for this jewelry store the number of customers in any given hour on a weekday is independent of the number of customers in any other hour on a weekday.

A prospective buyer of this jewelry store observes the business on a Wednesday for two one-hour periods (from 1 PM to 2 PM and 4 to 5 PM).

What is the probability that there will be 3 customers visiting from 1 PM to 2 PM and 5 customers visiting from 4 to 5 PM?

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Exam P Practice Problem 37 – Number of Claims

Problem 37A

An insurer issued insurance policies to two independent insureds (Insured #1 and Insured #2). The number of claims in the upcoming year for each insured has a Poisson distribution with mean 1.

  1. What is the probability that each of the insureds has at least one claim in the upcoming year?
  2. What is the probability that each of the insureds has at least one claim given that there is at least one claim in the upcoming year among these two insureds?

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Problem 37B

The number of ambulances passing through a busy intersection (while on duty) in a large city has a Poisson distribution with mean 0.2 per hour. Assume that the number of ambulances passing through this intersection in any given hour is independent of the number of ambulances in any other hour.

As part of a traffic study, this intersection is observed for two one-hour periods (from 9 AM to 10 AM and 1 PM to 2 PM).

  1. What is the probability that there is at least one ambulance passing through the intersection during these two hours?
  2. What is the probability that the period of 9 AM to 10 AM has at least one ambulance given that there is at least one ambulance passing through the intersection during these two hours?

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Exam P Practice Problem 36 – Number of Claims

This post has no alternate problem. It has one problem with 2 parts.

Problem 36

A claim examiner of an insurer reviews the claim history of two independent insureds. Let X be the annual number of claims of the first insured (Insured # 1). Let Y be the annual number of claims of the second insured (Insured # 2). The claim examiner finds tht X follows a Poisson distribution with mean 1, and that Y follows a distribution with the following probability function.

      \displaystyle P(Y=y)=\frac{3!}{y! (3-y)!} \ \biggl(\frac{1}{3}\biggr)^y \ \biggl(\frac{2}{3}\biggr)^{3-y} \ \ \ \ \ \ y=0,1,2,3
  1. Between these two insureds, what is the probability that one of the insureds has two more claims than the other insured in a year?
  2. Given that one of the insured has two more claims than the other insured, what is the probability that Insured # 1 has more claims than Insured # 2?

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Exam P Practice Problem 34 – Number of Claims

Problem 34A

An actuary is asked to review the claim history in a two-year period for one policyholder. The number of claims in any year for this policyholder has a Poisson distribution with mean 0.5 and is independent of the number of claims in any previous year.

The actuary finds that the policyholder has made 5 claims in the two-year period.

What is the probability that the policyholder has made more than three claims in the first year of the review period?

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Problem 34B

An actuary is asked to review the claim history in a two-year period for one policyholder. In the first year of the review period, the number of claims for this policyholder has a Poisson distribution with mean 0.3. In the second year, the number of claims for this policyholder has a Poisson distribution with mean 0.6.

The number of claims in one year is independent of the number of claims in any previous year.

The actuary finds that the policyholder has made 5 claims in the review period. What is the probability that the policyholder has made at least three claims in the first year of the review period?

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