Exam P Practice Problem 95 – Measuring Dispersion
Problem 95A
The lifetime (in years) of a machine for a manufacturing plant is modeled by the random variable . The following is the density function of .
Calculate the standard deviation of the lifetime of such a machine.
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Problem 95B
The travel time to work (in minutes) for an office worker has the following density function.
Calculate the variance of the travel time to work for this office worker.
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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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Exam P Practice Problem 83 – Claim Size of Auto Insurance Policies
Problem 83A
An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable where has a normal distribution with mean 0 and variance 1.5.
What is the expected claim size for such an auto insurance policy?
Problem 83B
An insurance company has a block of auto insurance policies. The claim size (in thousands) for a policy in this block of auto insurance policies is modeled by the random variable where has a normal distribution with mean 0 and variance 3.
What is the standard deviation of the claim size for such an auto insurance policy?
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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 71 – Estimating Claim Frequency
Problem 71A
An auto insurer issued policies to a large group of drivers under the age of 40. These drivers are classified into five distinct groups by age. These groups are equal in size.
The annual claim count distribution for any driver being insured by this insurer is assumed to be a binomial distribution. The following table shows more information about these drivers.
An insured driver is randomly selected from this large pool of insured and is observed to have one claim in the last year.
What is the probability that the mean number of claims in a year for this insured driver is greater than 1.5?
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Problem 71B
An auto insurer issued policies to a large group of drivers under the age of 40. These drivers are classified into five distinct groups by age. These groups are equal in size.
The annual claim count distribution for any driver being insured by this insurer is assumed to be a geometric distribution. The following table shows more information about these drivers.
An insured driver is randomly selected from this large pool of insured and is observed to have one claim in the last year.
What is the probability that the mean number of claims in a year for this insured driver is greater than 2.5?
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Exam P Practice Problem 58 – Dental Care and Vision Care Expenses
Problem 58A
A health plan offers dental care and vision care benefits. Let represents the total annual amount (in millions) paid in dental care benefits. Let represents the total annual amount (in millions) paid in vision care benefits.
The health plan determined that
 where follows a normal distribution with mean 0 and variance 1,
 where follows a normal distribution with mean 0 and variance 2, and
 and 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?
Problem 58B
A health plan offers dental care and vision care benefits. Let represents the total annual amount (in millions) paid in dental care benefits. Let represents the total annual amount (in millions) paid in vision care benefits.
The health plan determined that
 where follows a normal distribution with mean 0 and variance 1,
 where follows a normal distribution with mean 0 and variance 1, and
 and 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?
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Exam P Practice Problem 57 – Lifetimes of Machines
Problem 57A
A factory owner purchased two identical machines for her factory. Let and be the lifetimes (in years) of these two machines. These lifetimes are modeled by the following joint probability density function.
The machine whose lifetime is modeled by the random variable came online 2 years after the beginning of operation of the machine that is modeled by the random variable .
Given that exceeds 2, that is the probability that exceeds 3?
Problem 57B
A company purchased two machines for its factory. Let and be the lifetimes (in years) of these machines. The following is the joint density function of their lifetimes.
The machine whose lifetime is modeled by the random variable came online after the failure of the machine whose lifetime is modeled by .
What is the variance of the total time of operation of these two machines?
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Exam P Practice Problem 52 – Reliability of Refrigerators
Problem 52A
The time from initial purchase to the time of the first major repair (in years) for a brand of refrigerators is modeled by the random variable where is normally distributed with mean 1.2 and variance 2.25.
A customer just bought a brand new refrigerator of this particular brand. The refrigerator came with a twoyear warranty. During the warranty period, any repairs, both minor and major, are the responsibilities of the manufacturer.
What is the probability that the newly purchased refrigerator will not require major repairs during the warranty period?
Problem 52B
The time from initial purchase to the time of the first major repair (in years) for a brand of refrigerators is modeled by the random variable where is normally distributed with mean 0.8 and standard deviation 1.5.
What is the median length of time (from initial purchase) that is free of any need for major repairs?
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Exam P Practice Problem 46 – Finding Moment of a Sum
Problem 46A
Suppose that and are random losses that are jointly distributed with the following density function:
Find the second moment of the sum of the two losses.
Problem 46B
Suppose that and are random losses that are jointly distributed with the following density function:
Find the second moment of the sum of the two losses.
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