Tag Archives: Independent Random Varaibles

Exam P Practice Problem 76 – Quantifying Average Random Loss

Both Problem 76-A and Problem 76-B use the following information.

A property owner faces a series of independent random losses. Each loss is either 10 (with probability 0.4) or 50 (with probability 0.6).

Three such random losses are selected.

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Problem 76-A

What is the probability that the mean of the three losses is less than 30?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.06

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.29

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.35

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.43

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Problem 76-B

What is the expected value of the mean of the three losses?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 30

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 32

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 34

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35

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\copyright \ 2013 \ \ \text{Dan Ma}

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Exam P Practice Problem 75 – Travel Time to Work By Train

Both Problem 75-A and Problem 75-B use the following information.

A worker travels to work by train 5 days a week (Monday to Friday). The length of a train ride (in minutes) to work follows a continuous uniform distribution from 10 to 40.

The lengths of the train ride across the days of the week are independent.

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Problem 75-A

What is the probability that the shortest train ride during a work week is between 15 and 20 minutes?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.025

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.039

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.045

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.053

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.064

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Problem 75-B

What is the expected value of the longest train ride during a work week?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 25.9

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 28.2

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 33.3

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 35.7

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\copyright \ 2013-2016 \ \ \text{Dan Ma}

Exam P Practice Problem 74 – Review of Auto Insurance Claims

Both Problem 74-A and Problem 74-B use the following information.

An insurer issued policies to cover a large number of automobiles. Claim amounts (in thousands) from these policies are independent and are modeled by a continuous uniform distribution on (0,10).

The insurer randomly selects five claims for review.

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Problem 74-A

What is the probability that the minimum claim amount is between 2 thousands and 6 thousands?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.31

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.32

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.33

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.34

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75

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Problem 74-B

What is the expected value of the maximum claim amount?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 5.0

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

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 7.6

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.3

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8.5

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\copyright \ 2013 \ \ \text{Dan Ma}

Exam P Practice Problem 73 – Wait Time at a Busy Restaurant

Both Problem 73-A and Problem 73-B use the following information.

A certain restaurant is very busy in the evening time during the weekend. When customers arrive, they typically have to wait for a table.

When a customer has to wait for a table, the wait time (in minutes) follows a distribution with the following density function.

    \displaystyle f(x)=\frac{1}{1800} \ x, \ \ \ \ \ \ \ \ \ 0<x<60

A customer plans to dine at this restaurant on five Saturday evenings during the next 3 months. Assume that the customer will have to wait for a table on each of these evenings.

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Problem 73-A

What is the probability that the minimum wait time for a table during the next 3 months for this customer will be more than half an hour?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.24

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.25

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.36

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.42

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.75

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Problem 73-B

What is the mean of the maximum wait time (in minutes) for a table during the next 3 months for this customer?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 40.0

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 50.5

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 51.4

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 54.5

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 55.4

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\copyright \ 2013 \ \ \text{Dan Ma}

Exam P Practice Problem 70 – Real Estate Sales Contest

Problem 70-A

A commercial real estate property company has three sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these three agents are exponentially distributed with means 10 days, 15 days and 20 days.

These three agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the three agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the probability that the winning sale will take place within one week?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.14

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.22

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.50

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.78

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.86

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Problem 70-B

A commercial real estate property company has four sales agents who are actively selling commercial real estate properties. The times (in days) to the next successful sale for these four agents are exponentially distributed with means 10 days, 15 days and 20 days and 30 days.

These four agents work independently. So the time to the next successful sale for one agent is independent of the time to the next successful sale for any of the other agents.

To spur sales, the company has a contest among the four agents. Each agent produces a sale. The award will go to the first agent producing the first sale.

What is the expected waiting time (in days) from the beginning of the contest to the occurrence of the winning sale?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 4

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 6

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 8

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 10

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Exam P Practice Problem 69 – More Large Claim Studies

Problem 69-A

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.05^8}{5040} \ x^7 \ e^{-0.05 \ x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 80th percentile of the claim size distribution.

In a random sample of 10 claims from this portfolio of insurance policies, what is the probability that more than two of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3020

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.3222

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6242

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6778

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6980

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Problem 69-B

The size of a claim (in thousands) arising from a large portfolio of property and casualty insurance policies is modeled by the random variable X. The following is the probability density function of X.

      \displaystyle f(x)=\frac{0.04^{11}}{10!} \ x^{10} \ e^{-0.04 x} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>0

The size of a claim is independent of the size of any other claim in this portfolio of insurance policies.

An actuary is hired to study the large claims arising from these insurance policies, in particular, any claim size greater than the 90th percentile of the claim size distribution.

In a random sample of 15 claims from this portfolio of insurance policies, what is the probability that two or more of the claims are considered large by this actuary?

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      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.184

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.267

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.451

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.733

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0.816

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Exam P Practice Problem 58 – Dental Care and Vision Care Expenses

Problem 58-A

A health plan offers dental care and vision care benefits. Let X represents the total annual amount (in millions) paid in dental care benefits. Let Y represents the total annual amount (in millions) paid in vision care benefits.

The health plan determined that

    • X=K^2 where K follows a normal distribution with mean 0 and variance 1,
    • Y=L^2 where L follows a normal distribution with mean 0 and variance 2, and
    • K and L are independent.

Given that the total annual vision care benefits paid by the health plan exceeds 2.5 millions, what is the probability that the total annual dental care benefits paid by the health plan exceeds 2 millions?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0228

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.0793

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1586

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.8416

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.9207

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Problem 58-B

A health plan offers dental care and vision care benefits. Let X represents the total annual amount (in millions) paid in dental care benefits. Let Y represents the total annual amount (in millions) paid in vision care benefits.

The health plan determined that

    • X=2.5 K^2 where K follows a normal distribution with mean 0 and variance 1,
    • Y=5 L^2 where L follows a normal distribution with mean 0 and variance 1, and
    • K and L are independent.

What is the probability that the total annual dental care benefits exceeds 3 millions and that the total annual vision care benefits exceeds 4 millions?

      \displaystyle (A) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.1013

      \displaystyle (B) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.4565

      \displaystyle (C) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.6266

      \displaystyle (D) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7286

      \displaystyle (E) \ \ \ \ \ \ \ \ \ \ \ \ \ 0.7881

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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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