# 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:

$\displaystyle f(t)=\frac{1}{10 \ \sqrt{t}} \ \ e^{\displaystyle -\frac{\sqrt{t}}{5}} \ \ \ \ \ \ \ 0

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?

$\displaystyle A \ \ \ \ \ \ 0.5313$

$\displaystyle B \ \ \ \ \ \ 0.5875$

$\displaystyle C \ \ \ \ \ \ 0.6394$

$\displaystyle D \ \ \ \ \ \ 0.8310$

$\displaystyle E \ \ \ \ \ \ 0.9043$

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Problem 35B

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

$\displaystyle f(t)=\frac{3}{32} \ \ t \ (4-t) \ \ \ \ \ \ \ 0

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?

$\displaystyle A \ \ \ \ \ \ \frac{11}{32}$

$\displaystyle B \ \ \ \ \ \ \frac{16}{32}$

$\displaystyle C \ \ \ \ \ \ \frac{22}{32}$

$\displaystyle D \ \ \ \ \ \ \frac{24}{32}$

$\displaystyle E \ \ \ \ \ \ \frac{27}{32}$

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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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# 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?

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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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# Exam P Practice Problem 32 – Covariance

Problem 32A

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

$\displaystyle f(x,y)=\frac{1}{24} \ (x+y) \ \ \ \ \ 0

Calculate the covariance of these two losses.

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Problem 32B

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

$\displaystyle f(x,y)=\frac{1}{8} \ (x+y) \ \ \ \ \ 0

Calculate the covariance of these two losses.

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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 $X$ be the time (in hours) of his flight from City A to City B. Let $Y$ be the time (in hours) of the bus ride. The following is the joint density function of $X$ and $Y$.

$\displaystyle f(x,y)=\frac{1}{16} \ x \ y \ \ \ \ \ \ \ 0

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

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Problem 31B

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

$\displaystyle f(x,y)=0.025 \ x \ e^{-0.2 \ y} \ \ \ \ \ \ \ 0

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

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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 $X$ and $Y$ with the following joint density function:

$\displaystyle f(x,y)=\frac{3}{16} x^2 \ \ \ \ \ \ \ \ 0

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

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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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# 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.

$\displaystyle \begin{bmatrix} \text{Loss Amount}&\text{ }&\text{Probability} \\\text{ }&\text{ }&\text{ } \\ 5,000&\text{ }&0.5 \\ 10,000&\text{ }&0.3 \\ 50,000&\text{ }&0.1 \\ 75,000&\text{ }&0.05 \\ 100,000&\text{ }&0.03 \\ 150,000&\text{ }&0.015 \\ 200,000&\text{ }&0.005 \end{bmatrix}$

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.

$A. \ \ \ \ \ \text{15,000}$
$B. \ \ \ \ \ \text{15,500}$
$C. \ \ \ \ \ \text{20,500}$
$D. \ \ \ \ \ \text{31,000}$
$E. \ \ \ \ \ \text{30,000}$

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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?

$A. \ \ \ \ \ \text{29,541.49}$
$B. \ \ \ \ \ \text{30450.18}$
$C. \ \ \ \ \ \text{31,480.15}$
$D. \ \ \ \ \ \text{35,597.34}$
$E. \ \ \ \ \ \text{38,749.19}$

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# Exam P Practice Problem 28 – Wait Time to See Doctor

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Problem 28A

In the previous years, the wait time (in minutes) for a patient to see a doctor at a large medical clinic followed a distribution with the following moment generating function:

$\displaystyle M(t)=\frac{4}{4-100t} \ \ \ \ \ \ \ \ \ \ 4-100t>0$

In the current year, the administrator of the clinic still believes that the wait time to see a doctor follows the same distribution except that the mean wait time has increased by 50% (due to the fact that a higher volume of patients is being served by the same number of medical doctors).

What is the probability that in the current year the wait time for a patient to see a doctor is less than 30 minutes?

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Problem 28B
Use the same information as in Problem 28A. What is the percent increase in the variance of the wait time to see a doctor between the current year and the previous years?

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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 $\displaystyle \frac{1}{4}$ 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?

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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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# 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, $X$, of the bus ride from home to office follows a uniform distribution from $0$ to $20$ minutes and the length of time, $Y$, of the bus ride from office back to home follows a uniform distribution from $0$ to $25$ 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 $20$ minutes.

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

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