# SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS

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1 SOCIETY OF ACTUARIES/CASUALTY ACTUARIAL SOCIETY EXAM P PROBABILITY EXAM P SAMPLE QUESTIONS Copyright 007 by the Society of Actuaries and the Casualty Actuarial Society Some of the questions in this study note are taken from past SOA/CAS examinations. P PRINTED IN U.S.A. Page of 83

2 . A survey of a group s viewing habits over the last year revealed the following information: (i) (ii) (iii) (iv) (v) (vi) (vii) 8% watched gymnastics 9% watched baseball 9% watched soccer 4% watched gymnastics and baseball % watched baseball and soccer 0% watched gymnastics and soccer 8% watched all three sports. Calculate the percentage of the group that watched none of the three sports during the last year. (A) 4 (B) 36 (C) 4 (D) 5 (E) 60 Page of 83

3 . The probability that a visit to a primary care physician s (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCP s office, 30% are referred to specialists and 40% require lab work. Determine the probability that a visit to a PCP s office results in both lab work and referral to a specialist. (A) 0.05 (B) 0. (C) 0.8 (D) 0.5 (E) You are given P[A B] = 0.7 and P[A B ] = 0.9. Determine P[A]. (A) 0. (B) 0.3 (C) 0.4 (D) 0.6 (E) 0.8 Page 3 of 83

4 4. An urn contains 0 balls: 4 red and 6 blue. A second urn contains 6 red balls and an unknown number of blue balls. A single ball is drawn from each urn. The probability that both balls are the same color is Calculate the number of blue balls in the second urn. (A) 4 (B) 0 (C) 4 (D) 44 (E) An auto insurance company has 0,000 policyholders. Each policyholder is classified as (i) (ii) (iii) young or old; male or female; and married or single. Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders can also be classified as 30 young males, 300 married males, and 400 young married persons. Finally, 600 of the policyholders are young married males. How many of the company s policyholders are young, female, and single? Page 4 of 83

5 (A) 80 (B) 43 (C) 486 (D) 880 (E) A public health researcher examines the medical records of a group of 937 men who died in 999 and discovers that 0 of the men died from causes related to heart disease. Moreover, 3 of the 937 men had at least one parent who suffered from heart disease, and, of these 3 men, 0 died from causes related to heart disease. Determine the probability that a man randomly selected from this group died of causes related to heart disease, given that neither of his parents suffered from heart disease. (A) 0.5 (B) 0.73 (C) 0.4 (D) 0.37 (E) 0.54 Page 5 of 83

6 7. An insurance company estimates that 40% of policyholders who have only an auto policy will renew next year and 60% of policyholders who have only a homeowners policy will renew next year. The company estimates that 80% of policyholders who have both an auto and a homeowners policy will renew at least one of those policies next year. Company records show that 65% of policyholders have an auto policy, 50% of policyholders have a homeowners policy, and 5% of policyholders have both an auto and a homeowners policy. Using the company s estimates, calculate the percentage of policyholders that will renew at least one policy next year. (A) 0 (B) 9 (C) 4 (D) 53 (E) 70 Page 6 of 83

7 8. Among a large group of patients recovering from shoulder injuries, it is found that % visit both a physical therapist and a chiropractor, whereas % visit neither of these. The probability that a patient visits a chiropractor exceeds by 0.4 the probability that a patient visits a physical therapist. Determine the probability that a randomly chosen member of this group visits a physical therapist. (A) 0.6 (B) 0.38 (C) 0.40 (D) 0.48 (E) An insurance company examines its pool of auto insurance customers and gathers the following information: (i) (ii) (iii) (iv) All customers insure at least one car. 70% of the customers insure more than one car. 0% of the customers insure a sports car. Of those customers who insure more than one car, 5% insure a sports car. Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car. Page 7 of 83

8 (A) 0.3 (B) 0. (C) 0.4 (D) 0.5 (E) An insurance company examines its pool of auto insurance customers and gathers the following information: (i) (ii) (iii) (iv) All customers insure at least one car. 64% of the customers insure more than one car. 0% of the customers insure a sports car. Of those customers who insure more than one car, 5% insure a sports car. What is the probability that a randomly selected customer insures exactly one car, and that car is not a sports car? (A) 0.6 (B) 0.9 (C) 0.6 (D) 0.9 (E) 0.3 Page 8 of 83

9 . An actuary studying the insurance preferences of automobile owners makes the following conclusions: (i) (ii) (iii) An automobile owner is twice as likely to purchase collision coverage as disability coverage. The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage. The probability that an automobile owner purchases both collision and disability coverages is 0.5. What is the probability that an automobile owner purchases neither collision nor disability coverage? (A) 0.8 (B) 0.33 (C) 0.48 (D) 0.67 (E) 0.8 Page 9 of 83

10 . A doctor is studying the relationship between blood pressure and heartbeat abnormalities in her patients. She tests a random sample of her patients and notes their blood pressures (high, low, or normal) and their heartbeats (regular or irregular). She finds that: (i) (ii) (iii) (iv) (v) 4% have high blood pressure. % have low blood pressure. 5% have an irregular heartbeat. Of those with an irregular heartbeat, one-third have high blood pressure. Of those with normal blood pressure, one-eighth have an irregular heartbeat. What portion of the patients selected have a regular heartbeat and low blood pressure? (A) % (B) 5% (C) 8% (D) 9% (E) 0% 3. An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of women. For each of the three factors, the probability is 0. that a woman in the population has only this risk factor (and no others). For any two of the three factors, the probability is 0. that she has exactly these two risk factors (but not the other). The probability that a woman has all three risk factors, given that she has A and B, is 3. Page 0 of 83

11 What is the probability that a woman has none of the three risk factors, given that she does not have risk factor A? (A) 0.80 (B) 0.3 (C) (D) (E) In modeling the number of claims filed by an individual under an automobile policy during a three-year period, an actuary makes the simplifying assumption that for all integers n 0, p n+ = n claims during the period. 5 p n, where p n represents the probability that the policyholder files Under this assumption, what is the probability that a policyholder files more than one claim during the period? (A) 0.04 (B) 0.6 (C) 0.0 (D) 0.80 (E) 0.96 Page of 83

12 5. An insurer offers a health plan to the employees of a large company. As part of this plan, the individual employees may choose exactly two of the supplementary coverages A, B, and C, or they may choose no supplementary coverage. The proportions of the company s employees that choose coverages A, B, and C are 5,, and, 4 3 respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage. (A) 0 (B) (C) (D) (E) Page of 83

13 6. An insurance company determines that N, the number of claims received in a week, is a random variable with P[N = n] =, where n 0. The company also determines that n+ the number of claims received in a given week is independent of the number of claims received in any other week. Determine the probability that exactly seven claims will be received during a given two-week period. (A) (B) (C) (D) (E) Page 3 of 83

14 7. An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 5% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims. Calculate the probability that a claim submitted to the insurance company includes operating room charges. (A) 0.0 (B) 0.0 (C) 0.5 (D) 0.40 (E) Two instruments are used to measure the height, h, of a tower. The error made by the less accurate instrument is normally distributed with mean 0 and standard deviation h. The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation h. Assuming the two measurements are independent random variables, what is the probability that their average value is within 0.005h of the height of the tower? Page 4 of 83

15 (A) 0.38 (B) 0.47 (C) 0.68 (D) 0.84 (E) An auto insurance company insures drivers of all ages. An actuary compiled the following statistics on the company s insured drivers: Age of Driver Probability of Accident Portion of Company s Insured Drivers A randomly selected driver that the company insures has an accident. Calculate the probability that the driver was age 6-0. (A) 0.3 (B) 0.6 (C) 0.9 (D) 0.3 (E) 0.40 Page 5 of 83

16 0. An insurance company issues life insurance policies in three separate categories: standard, preferred, and ultra-preferred. Of the company s policyholders, 50% are standard, 40% are preferred, and 0% are ultra-preferred. Each standard policyholder has probability 0.00 of dying in the next year, each preferred policyholder has probability of dying in the next year, and each ultra-preferred policyholder has probability 0.00 of dying in the next year. A policyholder dies in the next year. What is the probability that the deceased policyholder was ultra-preferred? (A) (B) (C) (D) 0.04 (E) 0.87 Page 6 of 83

17 . Upon arrival at a hospital s emergency room, patients are categorized according to their condition as critical, serious, or stable. In the past year: (i) (ii) (iii) (iv) (vi) (vii) 0% of the emergency room patients were critical; 30% of the emergency room patients were serious; the rest of the emergency room patients were stable; 40% of the critical patients died; 0% of the serious patients died; and % of the stable patients died. Given that a patient survived, what is the probability that the patient was categorized as serious upon arrival? (A) 0.06 (B) 0.9 (C) 0.30 (D) 0.39 (E) A health study tracked a group of persons for five years. At the beginning of the study, 0% were classified as heavy smokers, 30% as light smokers, and 50% as nonsmokers. Results of the study showed that light smokers were twice as likely as nonsmokers to die during the five-year study, but only half as likely as heavy smokers. A randomly selected participant from the study died over the five-year period. Page 7 of 83

18 Calculate the probability that the participant was a heavy smoker. (A) 0.0 (B) 0.5 (C) 0.35 (D) 0.4 (E) An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any one-year period. The results of the study are presented below. Type of driver Percentage of all drivers Teen 8% Young adult 6% Midlife 45% Senior 3% Total 00% Probability of at least one collision Given that a driver has been involved in at least one collision in the past year, what is the probability that the driver is a young adult driver? (A) 0.06 (B) 0.6 (C) 0.9 (D) 0. (E) 0.5 Page 8 of 83

19 4. The number of injury claims per month is modeled by a random variable N with P[N = n] = ( n + )( n + ), where n 0. Determine the probability of at least one claim during a particular month, given that there have been at most four claims during that month. (A) (B) (C) (D) (E) A blood test indicates the presence of a particular disease 95% of the time when the disease is actually present. The same test indicates the presence of the disease 0.5% of the time when the disease is not present. One percent of the population actually has the disease. Calculate the probability that a person has the disease given that the test indicates the presence of the disease. Page 9 of 83

20 (A) 0.34 (B) (C) (D) (E) The probability that a randomly chosen male has a circulation problem is 0.5. Males who have a circulation problem are twice as likely to be smokers as those who do not have a circulation problem. What is the conditional probability that a male has a circulation problem, given that he is a smoker? (A) (B) (C) (D) (E) Page 0 of 83

21 7. A study of automobile accidents produced the following data: Model year Proportion of all vehicles Probability of involvement in an accident Other An automobile from one of the model years 997, 998, and 999 was involved in an accident. Determine the probability that the model year of this automobile is 997. (A) 0. (B) 0.30 (C) 0.33 (D) 0.45 (E) A hospital receives /5 of its flu vaccine shipments from Company X and the remainder of its shipments from other companies. Each shipment contains a very large number of vaccine vials. Page of 83

22 For Company X s shipments, 0% of the vials are ineffective. For every other company, % of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment and finds that one vial is ineffective. What is the probability that this shipment came from Company X? (A) 0.0 (B) 0.4 (C) 0.37 (D) 0.63 (E) The number of days that elapse between the beginning of a calendar year and the moment a high-risk driver is involved in an accident is exponentially distributed. An insurance company expects that 30% of high-risk drivers will be involved in an accident during the first 50 days of a calendar year. What portion of high-risk drivers are expected to be involved in an accident during the first 80 days of a calendar year? (A) 0.5 (B) 0.34 (C) 0.43 (D) 0.57 (E) 0.66 Page of 83

23 30. An actuary has discovered that policyholders are three times as likely to file two claims as to file four claims. If the number of claims filed has a Poisson distribution, what is the variance of the number of claims filed? (A) 3 (B) (C) (D) (E) 4 3. A company establishes a fund of 0 from which it wants to pay an amount, C, to any of its 0 employees who achieve a high performance level during the coming year. Each employee has a % chance of achieving a high performance level during the coming year, independent of any other employee. Determine the maximum value of C for which the probability is less than % that the fund will be inadequate to cover all payments for high performance. (A) 4 (B) 30 (C) 40 (D) 60 (E) 0 Page 3 of 83

24 3. A large pool of adults earning their first driver s license includes 50% low-risk drivers, 30% moderate-risk drivers, and 0% high-risk drivers. Because these drivers have no prior driving record, an insurance company considers each driver to be randomly selected from the pool. This month, the insurance company writes 4 new policies for adults earning their first driver s license. What is the probability that these 4 will contain at least two more high-risk drivers than low-risk drivers? (A) (B) 0.0 (C) 0.08 (D) (E) Page 4 of 83

25 33. The loss due to a fire in a commercial building is modeled by a random variable X with density function f( x) = 0.005(0 x) for 0 < x< 0 0 otherwise. Given that a fire loss exceeds 8, what is the probability that it exceeds 6? (A) (B) (C) (D) (E) The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function f, where f (x) is proportional to (0 + x). Calculate the probability that the lifetime of the machine part is less than 6. (A) 0.04 (B) 0.5 (C) 0.47 (D) 0.53 (E) 0.94 Page 5 of 83

26 35. The lifetime of a machine part has a continuous distribution on the interval (0, 40) with probability density function f, where f (x) is proportional to (0 + x). What is the probability that the lifetime of the machine part is less than 5? (A) 0.03 (B) 0.3 (C) 0.4 (D) 0.58 (E) A group insurance policy covers the medical claims of the employees of a small company. The value, V, of the claims made in one year is described by V = 00,000Y where Y is a random variable with density function k y y f( y) = 0 otherwise, 4 ( ) for 0 < < where k is a constant. What is the conditional probability that V exceeds 40,000, given that V exceeds 0,000? (A) 0.08 (B) 0.3 (C) 0.7 (D) 0.0 (E) 0.5 Page 6 of 83

27 37. The lifetime of a printer costing 00 is exponentially distributed with mean years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 00 printers, how much should it expect to pay in refunds? (A) 6,3 (B) 7,358 (C) 7,869 (D) 0,56 (E), An insurance company insures a large number of homes. The insured value, X, of a randomly selected home is assumed to follow a distribution with density function 4 3x for x> f( x) = 0 otherwise. Given that a randomly selected home is insured for at least.5, what is the probability that it is insured for less than? (A) (B) (C) (D) 0.89 (E) Page 7 of 83

28 39. A company prices its hurricane insurance using the following assumptions: (i) In any calendar year, there can be at most one hurricane. (ii) In any calendar year, the probability of a hurricane is (iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in any other calendar year. Using the company s assumptions, calculate the probability that there are fewer than 3 hurricanes in a 0-year period. (A) 0.06 (B) 0.9 (C) 0.38 (D) 0.6 (E) An insurance policy pays for a random loss X subject to a deductible of C, where 0< C < density function. The loss amount is modeled as a continuous random variable with f ( x) x for 0 < x < = 0 otherwise. Given a random loss X, the probability that the insurance payment is less than 0.5 is equal to Page 8 of 83

29 Calculate C. (A) 0. (B) 0.3 (C) 0.4 (D) 0.6 (E) A study is being conducted in which the health of two independent groups of ten policyholders is being monitored over a one-year period of time. Individual participants in the study drop out before the end of the study with probability 0. (independently of the other participants). What is the probability that at least 9 participants complete the study in one of the two groups, but not in both groups? (A) (B) 0.9 (C) 0.35 (D) (E) Page 9 of 83

30 4. For Company A there is a 60% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 0,000 and standard deviation,000. For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean 9,000 and standard deviation,000. Assume that the total claim amounts of the two companies are independent. What is the probability that, in the coming year, Company B s total claim amount will exceed Company A s total claim amount? (A) 0.80 (B) 0.85 (C) 0.7 (D) 0.3 (E) 0.40 Page 30 of 83

31 43. A company takes out an insurance policy to cover accidents that occur at its manufacturing plant. The probability that one or more accidents will occur during any given month is 3 5. The number of accidents that occur in any given month is independent of the number of accidents that occur in all other months. Calculate the probability that there will be at least four months in which no accidents occur before the fourth month in which at least one accident occurs. (A) 0.0 (B) 0. (C) 0.3 (D) 0.9 (E) An insurance policy pays 00 per day for up to 3 days of hospitalization and 50 per day for each day of hospitalization thereafter. The number of days of hospitalization, X, is a discrete random variable with probability function 6 k for k =,,3, 4,5 PX ( = k) = 5 0 otherwise. Determine the expected payment for hospitalization under this policy. Page 3 of 83

32 (A) 3 (B) 0 (C) 0 (D) 70 (E) Let X be a continuous random variable with density function f ( x) x for x 4 = 0 0 otherwise. Calculate the expected value of X. (A) (B) (C) (D) (E) Page 3 of 83

33 46. A device that continuously measures and records seismic activity is placed in a remote region. The time, T, to failure of this device is exponentially distributed with mean 3 years. Since the device will not be monitored during its first two years of service, the time to discovery of its failure is X = max(t, ). Determine E[X]. (A) + e 3 6 /3 4/3 (B) e + 5e (C) 3 (D) + 3 e /3 (E) A piece of equipment is being insured against early failure. The time from purchase until failure of the equipment is exponentially distributed with mean 0 years. The insurance will pay an amount x if the equipment fails during the first year, and it will pay 0.5x if failure occurs during the second or third year. If failure occurs after the first three years, no payment will be made. At what level must x be set if the expected payment made under this insurance is to be 000? (A) 3858 (B) 4449 (C) 538 (D) 5644 (E) 735 Page 33 of 83

34 48. An insurance policy on an electrical device pays a benefit of 4000 if the device fails during the first year. The amount of the benefit decreases by 000 each successive year until it reaches 0. If the device has not failed by the beginning of any given year, the probability of failure during that year is 0.4. What is the expected benefit under this policy? (A) 34 (B) 400 (C) 500 (D) 667 (E) An insurance policy pays an individual 00 per day for up to 3 days of hospitalization and 5 per day for each day of hospitalization thereafter. The number of days of hospitalization, X, is a discrete random variable with probability function 6 k for k =,,3, 4,5 PX ( = k) = 5 0 otherwise. Calculate the expected payment for hospitalization under this policy. Page 34 of 83

35 (A) 85 (B) 63 (C) 68 (D) 3 (E) A company buys a policy to insure its revenue in the event of major snowstorms that shut down business. The policy pays nothing for the first such snowstorm of the year and 0,000 for each one thereafter, until the end of the year. The number of major snowstorms per year that shut down business is assumed to have a Poisson distribution with mean.5. What is the expected amount paid to the company under this policy during a one-year period? (A),769 (B) 5,000 (C) 7,3 (D) 8,347 (E) 0,578 Page 35 of 83

36 5. A manufacturer s annual losses follow a distribution with density function f( x).5(0.6).5 for x > = x 0 otherwise. To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of. What is the mean of the manufacturer s annual losses not paid by the insurance policy? (A) 0.84 (B) 0.88 (C) 0.93 (D) 0.95 (E) An insurance company sells a one-year automobile policy with a deductible of. The probability that the insured will incur a loss is If there is a loss, the probability of a loss of amount N is K N, for N =,..., 5 and K a constant. These are the only possible loss amounts and no more than one loss can occur. Determine the net premium for this policy. Page 36 of 83

37 (A) 0.03 (B) (C) 0.07 (D) 0.0 (E) An insurance policy reimburses a loss up to a benefit limit of 0. The policyholder s loss, Y, follows a distribution with density function: for y > 3 f( y) = y 0, otherwise. What is the expected value of the benefit paid under the insurance policy? (A).0 (B).3 (C).8 (D).9 (E).0 Page 37 of 83

38 54. An auto insurance company insures an automobile worth 5,000 for one year under a policy with a,000 deductible. During the policy year there is a 0.04 chance of partial damage to the car and a 0.0 chance of a total loss of the car. If there is partial damage to the car, the amount X of damage (in thousands) follows a distribution with density function f( x) = 0 otherwise. x / e for 0 < x< 5 What is the expected claim payment? (A) 30 (B) 38 (C) 35 (D) 380 (E) An insurance company s monthly claims are modeled by a continuous, positive random variable X, whose probability density function is proportional to ( + x) 4, where 0 < x <. Determine the company s expected monthly claims. (A) (B) (C) 6 3 (D) (E) 3 Page 38 of 83

39 56. An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0, 000]. At what level must a deductible be set in order for the expected payment to be 5% of what it would be with no deductible? (A) 50 (B) 375 (C) 500 (D) 65 (E) An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generating function M X (t) = ( 500t) 4. Determine the standard deviation of the claim size for this class of accidents. (A),340 (B) 5,000 (C) 8,660 (D) 0,000 (E),80 Page 39 of 83

40 58. A company insures homes in three cities, J, K, and L. Since sufficient distance separates the cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are: M J (t) = ( t) 3 M K (t) = ( t).5 M L (t) = ( t) 4.5 Let X represent the combined losses from the three cities. Calculate E(X 3 ). (A),30 (B),08 (C) 5,760 (D) 8,000 (E) 0,560 Page 40 of 83

41 59. An insurer's annual weather-related loss, X, is a random variable with density function f( x) ( ) for x > 00 = 3.5 x 0 other wise. Calculate the difference between the 30 th and 70 th percentiles of X. (A) 35 (B) 93 (C) 4 (D) 3 (E) A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 00 with a variance of 60. If a tax of 0% is introduced on all items associated with the maintenance and repair of cars (i.e., everything is made 0% more expensive), what will be the variance of the annual cost of maintaining and repairing a car? (A) 08 (B) 60 (C) 70 (D) 3 (E) 374 Page 4 of 83

42 6. An insurer's annual weather-related loss, X, is a random variable with density function f( x) ( ) for x > 00 = 3.5 x 0 other wise. Calculate the difference between the 5 th and 75 th percentiles of X. (A) 4 (B) 48 (C) 67 (D) 4 (E) A random variable X has the cumulative distribution function ( ) F x Calculate the variance of X. 0 for x < x x+ = for x< for x. (A) (B) (C) (D) (E) Page 4 of 83

43 63. The warranty on a machine specifies that it will be replaced at failure or age 4, whichever occurs first. The machine s age at failure, X, has density function for 0 < x < 5 f( x) = 5 0 otherwise. Let Y be the age of the machine at the time of replacement. Determine the variance of Y. (A).3 (B).4 (C).7 (D). (E) A probability distribution of the claim sizes for an auto insurance policy is given in the table below: Claim Size Probability What percentage of the claims are within one standard deviation of the mean claim size? Page 43 of 83

44 (A) 45% (B) 55% (C) 68% (D) 85% (E) 00% 65. The owner of an automobile insures it against damage by purchasing an insurance policy with a deductible of 50. In the event that the automobile is damaged, repair costs can be modeled by a uniform random variable on the interval (0, 500). Determine the standard deviation of the insurance payment in the event that the automobile is damaged. (A) 36 (B) 403 (C) 433 (D) 464 (E) 5 Page 44 of 83

45 66. A company agrees to accept the highest of four sealed bids on a property. The four bids are regarded as four independent random variables with common cumulative distribution function 3 5 F( x) = ( + sin π x) for x. Which of the following represents the expected value of the accepted bid? (A) π 5/ 3/ x cos πxdx 6 5/ (B) ( + sin ) 6 3/ 5/ 4 π x dx (C) ( + sinπ ) 4 3/ 5/ x x dx (D) π cosπ ( + sinπ ) 4 3/ 5/ 4 x x dx (E) π x cosπx( + sinπx) 3/ 3 3 dx Page 45 of 83

46 67. A baseball team has scheduled its opening game for April. If it rains on April, the game is postponed and will be played on the next day that it does not rain. The team purchases insurance against rain. The policy will pay 000 for each day, up to days, that the opening game is postponed. The insurance company determines that the number of consecutive days of rain beginning on April is a Poisson random variable with mean 0.6. What is the standard deviation of the amount the insurance company will have to pay? (A) 668 (B) 699 (C) 775 (D) 87 (E) An insurance policy reimburses dental expense, X, up to a maximum benefit of 50. The probability density function for X is: where c is a constant x c e for x 0 f( x) = 0 otherwise, Calculate the median benefit for this policy. Page 46 of 83

47 (A) 6 (B) 65 (C) 73 (D) 8 (E) The time to failure of a component in an electronic device has an exponential distribution with a median of four hours. Calculate the probability that the component will work without failing for at least five hours. (A) 0.07 (B) 0.9 (C) 0.38 (D) 0.4 (E) 0.57 Page 47 of 83

48 70. An insurance company sells an auto insurance policy that covers losses incurred by a policyholder, subject to a deductible of 00. Losses incurred follow an exponential distribution with mean 300. What is the 95 th percentile of actual losses that exceed the deductible? (A) 600 (B) 700 (C) 800 (D) 900 (E) The time, T, that a manufacturing system is out of operation has cumulative distribution function fort Ft () > = t 0 otherwise. The resulting cost to the company is Y = T. Determine the density function of Y, for y > 4. Page 48 of 83

49 4 (A) y 8 (B) 3/ y 8 (C) 3 y (D) 6 y 04 (E) 5 y 7. An investment account earns an annual interest rate R that follows a uniform distribution on the interval ( 0.04, 0.08). The value of a 0,000 initial investment in this account after one year is given by V = 0,000e R. Determine the cumulative distribution function, F( v ), of V for values of v that satisfy ( ) 0< F v <. (A) (B) (C) (D) v/0,000 0,000e 0, v/0,000 5e 0.04 v 0, 408 0,833 0, v v (E) 5 ln , 000 Page 49 of 83

50 73. An actuary models the lifetime of a device using the random variable Y = 0X 0.8, where X is an exponential random variable with mean year. Determine the probability density function f (y), for y > 0, of the random variable Y. (A) (B) (C) (D) (E) 0y 8y 8y y 0. e 0. 0 y0.8 e 0. (0. y) e.5 (0. y).5 0.5(0. y) e (0. y) 0.5 (0. y) e Let T denote the time in minutes for a customer service representative to respond to 0 telephone inquiries. T is uniformly distributed on the interval with endpoints 8 minutes and minutes. Let R denote the average rate, in customers per minute, at which the representative responds to inquiries. Which of the following is the density function of the random variable R on the interval F0 0 r I? 8 H K (A) 5 (B) 5 3 r (C) 5 r 3r ln( ) Page 50 of 83

51 (D) (E) 0 r 5 r 75. The monthly profit of Company I can be modeled by a continuous random variable with density function f. Company II has a monthly profit that is twice that of Company I. Determine the probability density function of the monthly profit of Company II. (A) (B) x f x f x (C) f (D) f ( x ) (E) f ( x ) 76. Claim amounts for wind damage to insured homes are independent random variables with common density function 3 for x > f 4 ( x) = x 0 otherwise where x is the amount of a claim in thousands. Suppose 3 such claims will be made. Page 5 of 83

52 What is the expected value of the largest of the three claims? (A) 05 (B) 700 (C) 33 (D) 3375 (E) A device runs until either of two components fails, at which point the device stops running. The joint density function of the lifetimes of the two components, both measured in hours, is x+ y f( x, y) = for 0 < x< and 0 < y<. 8 What is the probability that the device fails during its first hour of operation? (A) 0.5 (B) 0.4 (C) 0.39 (D) 0.65 (E) Page 5 of 83

53 78. A device runs until either of two components fails, at which point the device stops running. The joint density function of the lifetimes of the two components, both measured in hours, is x+ y f( x, y) = for 0 < x< 3 and 0 < y< 3. 7 Calculate the probability that the device fails during its first hour of operation. (A) 0.04 (B) 0.4 (C) 0.44 (D) 0.59 (E) A device contains two components. The device fails if either component fails. The joint density function of the lifetimes of the components, measured in hours, is f ( st, ), where 0< s< and 0< t <. What is the probability that the device fails during the first half hour of operation? (A) (, ) (B) (, ) 0 0 f s t ds dt f s t ds dt (C) f (, ) s t ds dt (D) (, ) + (, ) f s t ds dt f s t ds dt Page 53 of 83

54 (E) (, ) + (, ) f st dsdt f st dsdt A charity receives 05 contributions. Contributions are assumed to be independent and identically distributed with mean 35 and standard deviation 50. Calculate the approximate 90 th percentile for the distribution of the total contributions received. (A) 6,38,000 (B) 6,338,000 (C) 6,343,000 (D) 6,784,000 (E) 6,977, Claims filed under auto insurance policies follow a normal distribution with mean 9,400 and standard deviation 5,000. What is the probability that the average of 5 randomly selected claims exceeds 0,000? (A) 0.0 (B) 0.5 (C) 0.7 (D) 0.33 (E) 0.45 Page 54 of 83

55 8. An insurance company issues 50 vision care insurance policies. The number of claims filed by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean. Assume the numbers of claims filed by distinct policyholders are independent of one another. What is the approximate probability that there is a total of between 450 and 600 claims during a one-year period? (A) 0.68 (B) 0.8 (C) 0.87 (D) 0.95 (E) A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance. A consumer buys a number of these bulbs with the intention of replacing them successively as they burn out. The light bulbs have independent lifetimes. What is the smallest number of bulbs to be purchased so that the succession of light bulbs produces light for at least 40 months with probability at least 0.977? (A) 4 (B) 6 (C) 0 (D) 40 (E) 55 Page 55 of 83

56 84. Let X and Y be the number of hours that a randomly selected person watches movies and sporting events, respectively, during a three-month period. The following information is known about X and Y: ( X ) ( Y ) ( X ) ( Y ) ( XY) E = 50 E = 0 Var = 50 Var = 30 Cov, = 0 One hundred people are randomly selected and observed for these three months. Let T be the total number of hours that these one hundred people watch movies or sporting events during this three-month period. Approximate the value of P(T < 700). (A) 0.6 (B) 0.84 (C) 0.87 (D) 0.9 (E) 0.97 Page 56 of 83

57 85. The total claim amount for a health insurance policy follows a distribution with density function ( x/000) f( x) = e for x > The premium for the policy is set at 00 over the expected total claim amount. If 00 policies are sold, what is the approximate probability that the insurance company will have claims exceeding the premiums collected? (A) 0.00 (B) 0.59 (C) (D) (E) A city has just added 00 new female recruits to its police force. The city will provide a pension to each new hire who remains with the force until retirement. In addition, if the new hire is married at the time of her retirement, a second pension will be provided for her husband. A consulting actuary makes the following assumptions: (i) (ii) Each new recruit has a 0.4 probability of remaining with the police force until retirement. Given that a new recruit reaches retirement with the police force, the probability that she is not married at the time of retirement is 0.5. Page 57 of 83

58 (iii) The number of pensions that the city will provide on behalf of each new hire is independent of the number of pensions it will provide on behalf of any other new hire. Determine the probability that the city will provide at most 90 pensions to the 00 new hires and their husbands. (A) 0.60 (B) 0.67 (C) 0.75 (D) 0.93 (E) In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from.5 years to.5 years. The healthcare data are based on a random sample of 48 people. What is the approximate probability that the mean of the rounded ages is within 0.5 years of the mean of the true ages? (A) 0.4 (B) 0.38 (C) 0.57 (D) 0.77 (E) 0.88 Page 58 of 83

59 88. The waiting time for the first claim from a good driver and the waiting time for the first claim from a bad driver are independent and follow exponential distributions with means 6 years and 3 years, respectively. What is the probability that the first claim from a good driver will be filed within 3 years and the first claim from a bad driver will be filed within years? (A) (B) d 3 76 e / e / + 8 e / i 76 e / 8 3 / / 76 / (C) e e + e 3 / / 3 / (D) e e + e (E) e e + e / / 76 / 89. The future lifetimes (in months) of two components of a machine have the following joint density function: 6 (50 x y) for 0 < x< 50 y< 50 f( x, y) = 5, otherwise. What is the probability that both components are still functioning 0 months from now? Page 59 of 83

60 (A) (B) (C) (D) (E) 6 5, , , , , x x y x x y 0 0 (50 x ydydx ) (50 x ydydx ) (50 x ydydx ) (50 x ydydx ) (50 x ydydx ) 90. An insurance company sells two types of auto insurance policies: Basic and Deluxe. The time until the next Basic Policy claim is an exponential random variable with mean two days. The time until the next Deluxe Policy claim is an independent exponential random variable with mean three days. What is the probability that the next claim will be a Deluxe Policy claim? (A) 0.7 (B) 0.3 (C) (D) (E) Page 60 of 83

61 9. An insurance company insures a large number of drivers. Let X be the random variable representing the company s losses under collision insurance, and let Y represent the company s losses under liability insurance. X and Y have joint density function x+ y for 0 < x< and 0 < y< f( x, y) = 4 0 otherwise. What is the probability that the total loss is at least? (A) 0.33 (B) 0.38 (C) 0.4 (D) 0.7 (E) Two insurers provide bids on an insurance policy to a large company. The bids must be between 000 and 00. The company decides to accept the lower bid if the two bids differ by 0 or more. Otherwise, the company will consider the two bids further. Assume that the two bids are independent and are both uniformly distributed on the interval from 000 to 00. Determine the probability that the company considers the two bids further. (A) 0.0 (B) 0.9 (C) 0.0 (D) 0.4 (E) 0.60 Page 6 of 83

62 93. A family buys two policies from the same insurance company. Losses under the two policies are independent and have continuous uniform distributions on the interval from 0 to 0. One policy has a deductible of and the other has a deductible of. The family experiences exactly one loss under each policy. Calculate the probability that the total benefit paid to the family does not exceed 5. (A) 0.3 (B) 0.5 (C) 0.30 (D) 0.3 (E) Let T be the time between a car accident and reporting a claim to the insurance company. Let T be the time between the report of the claim and payment of the claim. The joint density function of T and T, f ( t, t ), is constant over the region 0 < t < 6, 0< t < 6, t + t < 0, and zero otherwise. Determine E[T + T ], the expected time between a car accident and payment of the claim. (A) 4.9 (B) 5.0 (C) 5.7 (D) 6.0 (E) 6.7 Page 6 of 83

63 95. X and Y are independent random variables with common moment generating t / function M () t = e. Let W = X + Y and Z = Y X. Determine the joint moment generating function, ( ) M t, t, of W and Z. (A) e (B) e (C) e (D) e (E) e t + t ( t t ) ( t + t ) tt t + t 96. A tour operator has a bus that can accommodate 0 tourists. The operator knows that tourists may not show up, so he sells tickets. The probability that an individual tourist will not show up is 0.0, independent of all other tourists. Each ticket costs 50, and is non-refundable if a tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to pay 00 (ticket cost + 50 penalty) to the tourist. What is the expected revenue of the tour operator? Page 63 of 83

64 (A) 935 (B) 950 (C) 967 (D) 976 (E) Let T and T represent the lifetimes in hours of two linked components in an electronic device. The joint density function for T and T is uniform over the region defined by 0 t t L where L is a positive constant. Determine the expected value of the sum of the squares of T and T. (A) (B) (C) (D) L 3 L L 3 3L 4 (E) L Page 64 of 83

65 98. Let X, X, X 3 be a random sample from a discrete distribution with probability function for x = 0 3 px ( ) = for x = 3 0 otherwise Determine the moment generating function, M(t), of Y = X X X 3. (A) (B) (C) (D) (E) e e t t + e e 7 7 3t + e 3 3 t 3 3t 99. An insurance policy pays a total medical benefit consisting of two parts for each claim. Let X represent the part of the benefit that is paid to the surgeon, and let Y represent the part that is paid to the hospital. The variance of X is 5000, the variance of Y is 0,000, and the variance of the total benefit, X + Y, is 7,000. Due to increasing medical costs, the company that issues the policy decides to increase X by a flat amount of 00 per claim and to increase Y by 0% per claim. Page 65 of 83

66 Calculate the variance of the total benefit after these revisions have been made. (A) 8,00 (B) 8,800 (C) 9,300 (D) 9,50 (E) 0, A car dealership sells 0,, or luxury cars on any day. When selling a car, the dealer also tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold. P(X = 0, Y = 0) = 6 P(X =, Y = 0) = P(X =, Y = ) = 6 P(X =, Y = 0) = P(X =, Y = ) = 3 P(X =, Y = ) = 6 What is the variance of X? Page 66 of 83

67 (A) 0.47 (B) 0.58 (C) 0.83 (D).4 (E) The profit for a new product is given by Z = 3X Y 5. X and Y are independent random variables with Var(X) = and Var(Y) =. What is the variance of Z? (A) (B) 5 (C) 7 (D) (E) 6 Page 67 of 83

68 0. A company has two electric generators. The time until failure for each generator follows an exponential distribution with mean 0. The company will begin using the second generator immediately after the first one fails. What is the variance of the total time that the generators produce electricity? (A) 0 (B) 0 (C) 50 (D) 00 (E) In a small metropolitan area, annual losses due to storm, fire, and theft are assumed to be independent, exponentially distributed random variables with respective means.0,.5, and.4. Determine the probability that the maximum of these losses exceeds 3. (A) 0.00 (B) (C) 0.59 (D) 0.87 (E) 0.44 Page 68 of 83

69 04. A joint density function is given by where k is a constant. f ( x, y) = kx for 0 < x <, 0 < y < 0 otherwise, What is Cov(X,Y)? (A) 6 (B) 0 (C) (D) (E) Let X and Y be continuous random variables with joint density function 8 xy for 0 x, x y x f( x, y) = 3 0 otherwise. Calculate the covariance of X and Y. Page 69 of 83

70 (A) 0.04 (B) 0.5 (C) 0.67 (D) 0.80 (E) Let X and Y denote the values of two stocks at the end of a five-year period. X is uniformly distributed on the interval (0, ). Given X = x, Y is uniformly distributed on the interval (0, x). Determine Cov(X, Y) according to this model. (A) 0 (B) 4 (C) 6 (D) (E) 4 Page 70 of 83

71 07. Let X denote the size of a surgical claim and let Y denote the size of the associated hospital claim. An actuary is using a model in which E(X) = 5, E(X ) = 7.4, E(Y) = 7, E(Y ) = 5.4, and Var(X+Y) = 8. Let C = X+Y denote the size of the combined claims before the application of a 0% surcharge on the hospital portion of the claim, and let C denote the size of the combined claims after the application of that surcharge. Calculate Cov(C, C ). (A) 8.80 (B) 9.60 (C) 9.76 (D).5 (E) A device containing two key components fails when, and only when, both components fail. The lifetimes, T and T, of these components are independent with common density function f (t) = e t, t > 0. The cost, X, of operating the device until failure is T + T. Which of the following is the density function of X for x > 0? (A) e x/ e x / (B) ( x x e e ) (C) xe x Page 7 of 83

72 (D) (E) e x/ e x/ A company offers earthquake insurance. Annual premiums are modeled by an exponential random variable with mean. Annual claims are modeled by an exponential random variable with mean. Premiums and claims are independent. Let X denote the ratio of claims to premiums. What is the density function of X? (A) (B) (C) x + ( x + ) e x (D) e x (E) xe x 0. Let X and Y be continuous random variables with joint density function f ( x, y) 4 xy for 0 < x< and 0 < y< x = 0 otherwise. Page 7 of 83

73 Calculate P Y < X X = 3. (A) (B) (C) (D) (E) Once a fire is reported to a fire insurance company, the company makes an initial estimate, X, of the amount it will pay to the claimant for the fire loss. When the claim is finally settled, the company pays an amount, Y, to the claimant. The company has determined that X and Y have the joint density function (x ) ( x ) f ( x, y) = y x>, y >. x ( x ) Given that the initial claim estimated by the company is, determine the probability that the final settlement amount is between and 3. Page 73 of 83

74 (A) (B) (C) (D) (E) A company offers a basic life insurance policy to its employees, as well as a supplemental life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy. Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f(x,y) = (x + y) on the region where the density is positive. Given that 0% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy? (A) 0.00 (B) 0.03 (C) 0.08 (D) 0.47 (E) Page 74 of 83

75 3. Two life insurance policies, each with a death benefit of 0,000 and a one-time premium of 500, are sold to a couple, one for each person. The policies will expire at the end of the tenth year. The probability that only the wife will survive at least ten years is 0.05, the probability that only the husband will survive at least ten years is 0.0, and the probability that both of them will survive at least ten years is What is the expected excess of premiums over claims, given that the husband survives at least ten years? (A) 350 (B) 385 (C) 397 (D) 870 (E) A diagnostic test for the presence of a disease has two possible outcomes: for disease present and 0 for disease not present. Let X denote the disease state of a patient, and let Y denote the outcome of the diagnostic test. The joint probability function of X and Y is given by: P(X = 0, Y = 0) = P(X =, Y = 0) = P(X = 0, Y = ) = 0.05 P(X =, Y = ) = 0.5 Page 75 of 83

76 Calculate Var( YX= ). (A) 0.3 (B) 0.5 (C) 0.0 (D) 0.5 (E) The stock prices of two companies at the end of any given year are modeled with random variables X and Y that follow a distribution with joint density function f( x, y) = x for 0< x<, x< y< x+ 0 otherwise. What is the conditional variance of Y given that X = x? (A) (B) 7 6 (C) x + (D) x 6 (E) x + x + 3 Page 76 of 83

77 6. An actuary determines that the annual numbers of tornadoes in counties P and Q are jointly distributed as follows: Annual number of tornadoes in county Q 0 3 Annual number of tornadoes in county P Calculate the conditional variance of the annual number of tornadoes in county Q, given that there are no tornadoes in county P. (A) 0.5 (B) 0.84 (C) 0.88 (D) 0.99 (E) A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is [ ] 6 ( x + y) for x> 0, y > 0, x+ y< f( x, y) = 0 otherwise. Determine the probability that the portion of a claim representing damage to the house is less than 0.. Page 77 of 83

78 (A) (B) (C) (D) 0.5 (E) Let X and Y be continuous random variables with joint density function f ( x, y) Let g be the marginal density function of Y. Which of the following represents g? 5 y for x y x = 0 otherwise. (A) g( y) (B) g( y) (C) g( y) (D) g( y) (E) g( y) 5 y for 0 < y < = 0 otherwise 5y = for 0 otherwise x < y< x 5y for 0 < y < = 0 otherwise ( ) 5 for < = 0 otherwise 3/ / y y x y ( ) < = 0 otherwise < x 3/ / 5y y for 0 y< Page 78 of 83

79 9. An auto insurance policy will pay for damage to both the policyholder s car and the other driver s car in the event that the policyholder is responsible for an accident. The size of the payment for damage to the policyholder s car, X, has a marginal density function of for 0 < x <. Given X = x, the size of the payment for damage to the other driver s car, Y, has conditional density of for x < y < x +. If the policyholder is responsible for an accident, what is the probability that the payment for damage to the other driver s car will be greater than 0.500? (A) (B) (C) (D) (E) Page 79 of 83

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