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 63 – Total Minutes of Telephone Calls
Problem 63A
For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 12. The length in time (in minutes) of each telephone call has an exponential distribution with mean 5 minutes.
The length of time of one telephone call is independent of the length of time of any other telephone call.
On a given day, this individual makes or receives 4 telephone calls. What is the probability that this person is on the telephone for more than half an hour?
Problem 63B
For a certain individual, the daily number of telephone calls (incoming or outgoing) has a Poisson distribution with mean 16. The length in time (in minutes) of each telephone call has an exponential distribution with mean 8 minutes.
The length of time of one telephone call is independent of the length of time of any other telephone call.
On a given day, this individual makes or receives 5 telephone calls. What is the probability that this person is on the telephone for more than 45 minutes?
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Exam P Practice Problem 60 – Health Insurance Claim Frequency
Problem 60A
An insurance company issued health insurance policies to individuals. The company determined that , the number of claims filed by an insured in a year, is a random variable with the following probability function.
What is the probability that a random selected insured from this group of insured individuals will file more than 5 claims in a year?
Problem 60B
An insurance company issued health insurance policies to individuals. The company determined that , the number of claims filed by an insured in a year, is a random variable with the following probability function.
The number of claims filed by one insured individual is independent of the number of claims filed by any other insured individual.
An actuary studied three randomly selected insured individuals from this group of individuals who purchased health policies from this company. What is the probability that these three insured individuals will file more than 6 claims in a year?
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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 48 – Dental and Vision Expenses
Problem 48A
An insurance company sells an ancillary health benefit plan that reimburses dental expenses and vision care expenses to the plan members. The plan provides no other benefits in addition to dental and vision care.
For the basic plan, the annual amount of dental reimbursement and the annual amount of vision care reimbursement are identically and exponentially distributed with mean 2 (in hundreds).
An actuary is designing a deluxe ancillary plan that provides similar but richer benefits. The annual amount of deluxe dental reimbursement is four times that of the basic plan. The annual amount of deluxe vision care reimbursement is two times that of the basic plan. Except for the richer benefit amounts, the actuary believes that the deluxe plan reimbursements have the same underlying probability distribution as the basic plan.
For both the basic plan and deluxe plan, the annual amount of dental reimbursement is independent of the annual amount of vision care reimbursement.
Which of the following is the probability density function of the total annual amount of expenses reimbursed by the deluxe plan?
Problem 48B
When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area A has an exponential distribution with mean 20.
When tornadoes occur, the total annual amount of property damages due to tornadoes (in millions) in area B has an exponential distribution with mean 25.
Since area A and area B are sufficiently far apart, assume that the amount of tornado damages in one area is independent of the amount of damages in the other area.
What is the probability density function of the total annual amount of tornado damages for these two areas?
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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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Exam P Practice Problem 44 – Traffic Statistics Using Poisson Distribution
Problem 44A
The number of car accidents in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 4 per week. The number of car accidents in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8 per week.
Assume that on a weekly basis, the number of accidents in one highway is independent of the number of accidents in the other highway.
In one particular week, exactly 5 auto accidents took place in these two highways. What is the probability that Highway #1 had exactly 2 accidents in this particular week?
Problem 44B
The number of cars (in a day) that break down in a stretch of a highway (Highway #1) has a Poisson distribution with a mean of 16. The number of cars (in a day) that break down in a stretch of another highway (Highway #2) has a Poisson distribution with a mean of 8.
Assume that on a daily basis, the number of cars breaking down in one highway is independent of the number of cars breaking down in the other highway.
In one particular day, exactly 8 cars were found to break down in these two highways. What is the probability that Highway #1 had exactly 5 cars breaking down in this particular day?
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Exam P Practice Problem 43 – Joint Random Losses
Problem 43A
Two random losses and are jointly distributed according to the following density function:
Suppose that these two random losses had occurred. If the total loss is 5, what is the expected value of the loss ?
Problem 43B
Two random losses and are jointly distributed according to the following density function:
Suppose that these two random losses had occurred. If the total loss is 6, what is the expected value of the loss ?
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Exam P Practice Problem 41 – Conditional Expected Number of Balls
Problem 41A
An insurer has a block of business where the number of claims in a year for a policyholder in the block has the following probability distribution.
Two policyholders from this block are randomly selected and observed for a year. It is found that there are exactly four claims for these two policyholders in the past year. What is the probability that one of the policyholders has exactly three claims?
Problem 41B
There are two bowls containing red balls and white balls. Bowl 1 contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 1 without replacement. Bowl 2 also contains 5 red balls and 5 white balls. Five balls are randomly selected from Bowl 2 with replacement.
If the total number of red balls drawn from the two bowls is eight, what is the expected number of red balls from Bowl 2?
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Exam P Practice Problem 40 – Total Claim Amount
Problem 40A
The number of claims in a calendar year for an insured has a probability function indicated below.
When a claim occurs, the claim amount , regardless of how many claims the insured will have in the calendar year, has probabilities and . The claim amounts in a calendar year for this insured are independent.
Let be the total claim amount for this insured in a calendar year. Calculate .
Problem 40B
A bowl has 3 red balls and 6 white balls. Select two balls at random from this bowl with replacement. Let be the number of red balls found in the two selected balls. When where , roll a fair die times.
Let be the sum of the rolls of the die. Calculate .
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