Exam P Practice Problem 96 – Expected Insurance Payment
Problem 96A
An insurance policy is purchased to cover a random loss subject to a deductible of 1. The cumulative distribution function of the loss amount is:
Given a random loss , determine the expected payment made under this insurance policy?
Problem 96B
An insurance policy is purchased to cover a random loss subject to a deductible of 2. The density function of the loss amount is:
Given a random loss , what is the expected benefit paid by this insurance policy?
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Answers

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Exam P Practice Problem 88 – Expected Value of Insurance Payments
Problem 88A
A random loss has a uniform distribution over the interval . An insurance policy is purchased to reimburse the loss up to a maximum limit of where .
The expected value of the benefit payment under this policy is 8.4. Calculate the value of .
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Problem 88B
An individual purchases an insurance policy to cover a loss whose density function is:
The insurance policy reimburses the policy owner up to a benefit limit of 4 for each loss. What is the expected value of insurance payment made to the policy owner?
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Answers

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Exam P Practice Problem 86 – Finding Mean and Variance
The following is the cumulative distribution function of the random variable .
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Problem 86A
Calculate the expected value of .
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Problem 86B
Calculate the variance of .
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Answers

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Exam P Practice Problem 77 – Estimating Random Claim Sizes
Problem 77A
The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.
What is the probability that a randomly selected claim from this insurance policy is within 120% of the mean claim size?
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Problem 77B
The probability distribution of the claim size from an auto insurance policy randomly selected from a large pool of policies is described by the following density function.
What is the probability that a randomly selected claim from this insurance policy is within onehalf of a standard deviation of the mean claim size?
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Exam P Practice Problem 76 – Quantifying Average Random Loss
Both Problem 76A and Problem 76B 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 76A
What is the probability that the mean of the three losses is less than 30?
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Problem 76B
What is the expected value of the mean of the three losses?
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Exam P Practice Problem 75 – Travel Time to Work By Train
Both Problem 75A and Problem 75B 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 75A
What is the probability that the shortest train ride during a work week is between 15 and 20 minutes?
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Problem 75B
What is the expected value of the longest train ride during a work week?
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Exam P Practice Problem 74 – Review of Auto Insurance Claims
Both Problem 74A and Problem 74B 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 74A
What is the probability that the minimum claim amount is between 2 thousands and 6 thousands?
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Problem 74B
What is the expected value of the maximum claim amount?
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Exam P Practice Problem 73 – Wait Time at a Busy Restaurant
Both Problem 73A and Problem 73B 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.
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 73A
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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Problem 73B
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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Exam P Practice Problem 70 – Real Estate Sales Contest
Problem 70A
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?
Problem 70B
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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Exam P Practice Problem 68 – Large Claim Studies
Problem 68A
An insurance company has a large block of auto insurance policies such that the claim sizes (in thousands) from a policy in this block are modeled by the random variable . The following is the probability density function of .
An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over ten thousands is considered a large claim.
What is the mean size of the claims studied by this actuary?
Problem 68B
An insurance company has a large block of auto insurance policies such that the claim sizes (in thousands) from a policy in this block are modeled by the random variable . The following is the probability density function of .
An actuary is hired to study the large claims arising from these auto insurance policies. In this study, any claim size over five thousands is considered a large claim.
What is the mean size of the claims studied by this actuary?
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