# Exam P Practice Problem 35 – Lifetime of Machine

**Problem 35A**

The lifetime (in years) of a manufacturing equipment follows a distribution with the following probability density function:

A factory owner just bought such an equipment that is 5-year old and is in working condition. What is the probability that it will work for another 5 years?

**Problem 35B**

The time (in weeks) it takes a home builder to build a house follows a continuous distribution with the following density function:

A customer signs a contract to hire this home builder to build a house. The construction begins immediately after the signing of the contract.

Two weeks into the contract, the customer finds that the house is not completed. What is the probability that the total time from start to completion is three weeks or less?

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**Answers**

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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?

**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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**Answers**

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# Exam P Practice Problem 33 – Total Claim Amount

**Problem 33A**

The number of claims in a calendar year for an insured has a Poisson distribution with mean 1. When a claim occurs, the individual claim amount, regardless of how many claims the insured will have in the calendar year, is either 1 or 2, with probabilities 0.6 and 0.4, respectively.

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

In the next calendar year, what is the probability that the total claim amount for this insured will be 5?

**Problem 33B**

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, regardless of how many claims the insured will have in the calendar year, is either 2 or 4, with equal probabilities.

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

In the next calendar year, what is the probability that the total claim amount for this insured will be 10?

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**Answers**

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# Exam P Practice Problem 32 – Covariance

**Problem 32A**

Suppose that and are two random losses with the following joint density function:

Calculate the covariance of these two losses.

**Problem 32B**

Suppose that and are two random losses with the following joint density function:

Calculate the covariance of these two losses.

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**Answers**

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# Exam P Practice Problem 31 – Covariance

**Problem 31A**

A traveler is at the airport of City A and wants to reach his home in City B. He plans to take an airplane flight from City A to the airport of City B. Once he arrives in the airport of City B, he plans to reach his home by riding in a bus. Let be the time (in hours) of his flight from City A to City B. Let be the time (in hours) of the bus ride. The following is the joint density function of and .

Let be the total time of travel from the airport of City A to his home in City B. Calculate the covariance of and .

**Problem 31B**

An investor is facing two potential financial losses and with the following joint density function:

Let be the total of these two losses. Calculate the covariance of and .

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**Answers**

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# Exam P Practice Problem 30 – Mean and Variance of Total Loss

**Practice Problem 30A**

A property owner faces two random losses and with the following joint density function:

The property owner purchases an insurance policy to cover the total loss . This policy does not have a deductible. What is the expected payment to the property owner under this policy?

**Practice Problem 30B**

Use the same information as in Problem 30A. What is the variance of the payment to the property owner under this policy?

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**Answers**

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# Exam P Practice Problem 29 – Expected Insurance Payment

**Problem 29A**

When there is a loss due to fire damage to a house, the loss follows the distribution indicated below.

The owner of the house purchases an insurance policy with a deductible of 5,000 to insure againse this loss. What is the expected payment to the owner under this policy.

**Problem 29B**

Use the same information as in Problem 29A. What is the standard deviation of the payment to the owner under this insurance policy?

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**Answers**

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# Exam P Practice Problem 27 – Exponential Distributions

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**Problem 27A**

For a certain brand of light bulbs, the time (in thousands of hours) until the instant a light bulb burns out follows an exponential distribution. From past experience, we know that of the light bulbs will go out within 500 hours of use. What proportion of the light bulbs can be expected to go out within 1000 hours of use?

**Problem 27B**

In the emergency room of a large hospital in a metropolitan area, the wait time (in hours) to see a doctor follows an exponential distribution. From past experience, 40% of the wait times are 3 hours or less. What proportion of the patients in the emergency room can be expected to wait more than 15 hours to see a doctor?

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**Answers**

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# Exam P Practice Problem 26 – Uniform Distribution

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This post has no alternate problem. It has one problem with 3 parts.

**Problem 26**

For an office worker, the length of time, , of the bus ride from home to office follows a uniform distribution from to minutes and the length of time, , of the bus ride from office back to home follows a uniform distribution from to minutes.

Suppose that length of bus rides in one direction is independent of the length of bus rides in the other direction.

In recent weeks, the worker finds that the total daily time spent on bus rides always exceeds minutes.

- What is the probability that the bus ride to return home will take more than 10 minutes?
- What is the expected length of the bus rides from office to home?
- What is the variance of the length of the bus rides from office to home?

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