Category Archives: Independent Sum

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 60 – Health Insurance Claim Frequency

Problem 60-A

An insurance company issued health insurance policies to individuals. The company determined that Y, the number of claims filed by an insured in a year, is a random variable with the following probability function.

      \displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots

What is the probability that a random selected insured from this group of insured individuals will file more than 5 claims in a year?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0226

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0277

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0357

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0503

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0749

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Problem 60-B

An insurance company issued health insurance policies to individuals. The company determined that Y, the number of claims filed by an insured in a year, is a random variable with the following probability function.

      \displaystyle P(Y=y)=0.45 \ (0.55)^{\displaystyle y} \ \ \ \ \ \ y=0,1,2,3,\cdots

The number of claims filed by one insured individual is independent of the number of claims filed by any other insured individual.

An actuary studied three randomly selected insured individuals from this group of individuals who purchased health policies from this company. What is the probability that these three insured individuals will file more than 6 claims in a year?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0457

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0706

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1495

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2201

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.2406

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Exam P Practice Problem 57 – Lifetimes of Machines

Problem 57-A

A factory owner purchased two identical machines for her factory. Let X and Y be the lifetimes (in years) of these two machines. These lifetimes are modeled by the following joint probability density function.

      \displaystyle f(x,y)=\frac{0.01}{\sqrt{x} \ \sqrt{y}} \ e^{-0.2 \sqrt{x}} \ e^{-0.2 \sqrt{y}} \ \ \ \ \ \ \ 0<x<\infty, \ \ \ 0<y<\infty

The machine whose lifetime is modeled by the random variable Y came online 2 years after the beginning of operation of the machine that is modeled by the random variable X.

Given that X exceeds 2, that is the probability that Y exceeds 3?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 0.2928

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 0.4670

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 0.5330

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 0.7072

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 0.7536

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Problem 57-B

A company purchased two machines for its factory. Let X and Y be the lifetimes (in years) of these machines. The following is the joint density function of their lifetimes.

      \displaystyle f(x,y)=\frac{3}{125} \ y \ e^{-0.3 x} \ \ \ \ \ \ \ 0<x<\infty, \ \ \ 0<y<5

The machine whose lifetime is modeled by the random variable Y came online after the failure of the machine whose lifetime is modeled by X.

What is the variance of the total time of operation of these two machines?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ 12.50

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ 13.60

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ 17.20

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ 19.85

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ 23.61

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Exam P Practice Problem 48 – Dental and Vision Expenses

Problem 48-A

An insurance company sells an ancillary health benefit plan that reimburses dental expenses and vision care expenses to the plan members. The plan provides no other benefits in addition to dental and vision care.

For the basic plan, the annual amount of dental reimbursement and the annual amount of vision care reimbursement are identically and exponentially distributed with mean 2 (in hundreds).

An actuary is designing a deluxe ancillary plan that provides similar but richer benefits. The annual amount of deluxe dental reimbursement is four times that of the basic plan. The annual amount of deluxe vision care reimbursement is two times that of the basic plan. Except for the richer benefit amounts, the actuary believes that the deluxe plan reimbursements have the same underlying probability distribution as the basic plan.

For both the basic plan and deluxe plan, the annual amount of dental reimbursement is independent of the annual amount of vision care reimbursement.

Which of the following is the probability density function of the total annual amount of expenses reimbursed by the deluxe plan?

      \displaystyle (A) \ \ \ \ \ \ f(x)=\frac{1}{64 \times 5!} \ x^5 \ e^{ -0.5 \ x}

      \displaystyle (B) \ \ \ \ \ \ f(x)=4 \ e^{-0.125 \ x} - 4 \ e^{-0.25 \ x}

      \displaystyle (C) \ \ \ \ \ \ f(x)=0.125 \ e^{-0.125 \ x} + 0.25 \ e^{-0.25 \ x}

      \displaystyle (D) \ \ \ \ \ \ f(x)=0.25 \ e^{-0.125 \ x} - 0.25 \ e^{-0.25 \ x}

      \displaystyle (E) \ \ \ \ \ \ f(x)=1.25 \ e^{-0.25 \ x} - 0.5 \ e^{-0.125 \ x}

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Problem 48-B

When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area A has an exponential distribution with mean 20.

When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area B has an exponential distribution with mean 25.

Since area A and area B are sufficiently far apart, assume that the amount of tornado damages in one area is independent of the amount of damages in the other area.

What is the probability density function of the total annual amount of tornado damages for these two areas?

      \displaystyle (A) \ \ \ \ \ \ f(x)= e^{-0.04 \ x} - \ e^{-0.05 \ x}

      \displaystyle (B) \ \ \ \ \ \ f(x)=0.2 \ e^{-0.04 \ x} - 0.2 \ e^{-0.05 \ x}

      \displaystyle (C) \ \ \ \ \ \ f(x)=0.04 \ e^{-0.04 \ x} + 0.05 \ e^{-0.05 \ x}

      \displaystyle (D) \ \ \ \ \ \ f(x)=\frac{1}{45} \ e^{- \frac{1}{45} \ x}

      \displaystyle (E) \ \ \ \ \ \ f(x)=0.3 \ e^{-0.05 \ x} - 0.2 \ e^{-0.04 \ x}

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Exam P Practice Problem 46 – Finding Moment of a Sum

Problem 46-A

Suppose that X and Y are random losses that are jointly distributed with the following density function:

      \displaystyle f(x,y)=\frac{1}{40} \ (4-x) \ \ \ \ \ \ 0<x<4, \ \ 0<y<5

Find the second moment of the sum of the two losses.

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Problem 46-B

Suppose that X and Y are random losses that are jointly distributed with the following density function:

      \displaystyle f(x,y)=32 \ x^2 \ y \ e^{-2 \ (x \ + \ y)} \ \ \ \ \ \ 0<x<\infty, \ \ 0<y<\infty

Find the second moment of the sum of the two losses.

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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 43 – Joint Random Losses

Problem 43-A

Two random losses X and Y are jointly distributed according to the following density function:

      \displaystyle f(x,y)=\frac{1}{64} \ x \ y \ \ \ \ \ \ 0<x<4, \ 0<y<4

Suppose that these two random losses had occurred. If the total loss is 5, what is the expected value of the loss X?

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Problem 43-B

Two random losses X and Y are jointly distributed according to the following density function:

      \displaystyle f(x,y)=\frac{1}{64} \ (4-x) \ (4-y) \ \ \ \ \ \ 0<x<4, \ 0<y<4

Suppose that these two random losses had occurred. If the total loss is 6, what is the expected value of the loss X?

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Exam P Practice Problem 41 – Conditional Expected Number of Balls

Problem 41-A

An insurer has a block of business where the number of claims in a year for a policyholder in the block has the following probability distribution.

            \displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability}  \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle 0.08 \\\text{ }&\text{ }&\text{ }  \\ 1&\text{ }&\displaystyle 0.35 \\\text{ }&\text{ }&\text{ }  \\ 2&\text{ }&\displaystyle 0.40 \\\text{ }&\text{ }&\text{ }  \\ 3&\text{ }&\displaystyle 0.12 \\\text{ }&\text{ }&\text{ }  \\ 4&\text{ }&\displaystyle 0.0375 \\\text{ }&\text{ }&\text{ }  \\ 5&\text{ }&\displaystyle 0.0125   \end{bmatrix}

Two policyholders from this block are randomly selected and observed for a year. It is found that there are exactly four claims for these two policyholders in the past year. What is the probability that one of the policyholders has exactly three claims?

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Problem 41-B

There are two bowls containing red balls and white balls. Bowl 1 contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 1 without replacement. Bowl 2 also contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 2 with replacement.

If the total number of red balls drawn from the two bowls is eight, what is the expected number of red balls from Bowl 2?

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Exam P Practice Problem 40 – Total Claim Amount

Problem 40-A

The number of claims in a calendar year for an insured has a probability function indicated below.

            \displaystyle \begin{bmatrix} \text{Number of Claims}&\text{ }&\text{Probability}  \\\text{ }&\text{ }&\text{ } \\ 0&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ }  \\ 1&\text{ }&\displaystyle \frac{27}{64} \\\text{ }&\text{ }&\text{ }  \\ 2&\text{ }&\displaystyle \frac{9}{64} \\\text{ }&\text{ }&\text{ }  \\ 3&\text{ }&\displaystyle \frac{1}{64}     \end{bmatrix}

When a claim occurs, the claim amount X, regardless of how many claims the insured will have in the calendar year, has probabilities P(X=1)=0.8 and P(X=2)=0.2. The claim amounts in a calendar year for this insured are independent.

Let T be the total claim amount for this insured in a calendar year. Calculate P(3 \le T \le 4).

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Problem 40-B

A bowl has 3 red balls and 6 white balls. Select two balls at random from this bowl with replacement. Let N be the number of red balls found in the two selected balls. When N=n where n>0, roll a fair die n times.

Let W be the sum of the rolls of the die. Calculate P(4 \le W \le 5).

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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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