Actuarial Mathematics 1
Actuarial Mathematics 1 MAP 4172
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Alvera Ryan DVM
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This 43 page Class Notes was uploaded by Alvera Ryan DVM on Monday October 12, 2015. The Class Notes belongs to MAP 4172 at Florida Atlantic University taught by Philip Pina in Fall. Since its upload, it has received 27 views. For similar materials see /class/221653/map-4172-florida-atlantic-university in Applied Mathematics at Florida Atlantic University.
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Date Created: 10/12/15
EXAM SAMPLE QUESTIONS 1 A survey of a group s Viewing habits over the last year revealed the following information 3 CA3 i 28 watched gymnastics ii 29 watched baseball iii 19 watched soccer iv 14 watched gymnastics and baseball V 12 watched baseball and soccer Vi 10 watched gymnastics and soccer Vii 8 watched all three sports Calculate the percentage of the group that watched none of the three sports during the last year A B C D E 24 36 41 52 60 AAAAA VVVVV The probability that a Visit to a primary care physician s PCP of ce results in neither lab work nor referral to a specialist is 35 Of those coming to a PCPAfs o ice7 30 are referred to specialists and 40 require lab work Determine the probability that a Visit to a PCP s of ce results in both lab work and referral to a specialist A 005 B 012 c 018 025 035 D E You are given PM U B 07 and PAU Bl 09 Determine PA A B C D E 02 03 04 06 08 AAAAA VVVVV An urn contains 10 balls 4 red and 6 blue A second urn contains 16 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 044 Calculate the number of blue balls in the second urn OT A Bi V a q An auto insurance company has 10000 policyholders Each policyholder is classi ed as i young or old ii male or female and iii married or single Of these policyholders 3000 are young 4600 are male and 7000 are married The policyholders can also be classi ed as 1320 young males 3010 married males and 1400 young married persons Finally 600 of the policyholders are young married males How many of the companyAfs policyholders are young female and single A B 280 C 423 486 880 896 D E A public health researcher examines the medical records of a group of 937 men who died in 1999 and discovers that 210 of the men died from causes related to heart disease Moreover 312 of the 937 men had at least one parent who suffered from heart disease and of these 312 men 102 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 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 15 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 B C D E 20 29 41 53 70 AAAAA VVVVV Among a large group of patients recovering from shoulder injuries it is found that 22 visit both a physical therapist and a chiropractor whereas 12 visit neither of these The probability that a patient visits a chii ropractor exceeds by 014 the probability that a patient visits a physical therapist Determine the probability that a randomly chosen member of this group visits a physical therapist 3 A B 026 038 040 O D E 048 062 An insurance company examines its pool of auto insurance customers and gathers the following information i All customers insure at least one car ii 70 of the customers insure more than one car iii 20 of the customers insure a sports car iv Of those customers who insure more than one car7 15 insure a sports car Calculate the probability that a randomly selected customer insures exactly one car and that car is not a sports car A B 013 C 021 024 025 030 D E An insurance company examines its pool of auto insurance customers and gathers the following information i All customers insure at least one car ii 64 of the customers insure more than one car iii 20 of the customers insure a sports car iv Of those customers who insure more than one car7 15 insure a sports car What is the probability that a randomly selected customer insures exactly one car7 and that car is not a sports car A B 016 C 019 026 029 031 U E An actuary studying the insurance preferences of automobile owners makes the following conclusions i An automobile owner is twice as likely to purchase collision coverage as disability coverage ii The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage iii The probability that an automobile owner purchases both collision and disability coverages is 015 What is the probability that an automobile owner purchases neither collision nor disability coverage A 018 H 3 B 033 c 048 067 082 D E 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 nds that i 14 have high blood pressure ii 22 have low blood pressure iii 15 have an irregular heartbeat iv Of those with an irregular heartbeat oneethird have high blood pressure v Of those with normal blood pressure oneeeighth have an irregular heartbeat What portion of the patients selected have a regular heartbeat and low blood pressure 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 01 that a woman in the population has only this risk factor and no others For any two of the three factors the probability is 012 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 1 A dB39i an 1s3 What is the probability that a woman has none of the three risk factors given that she does not have risk factor In modeling the number of claims led by an individual under an automobile policy during a threeeyear period an actuary makes the simplifying assumption that for all integers n 2 0 pn1 l where 13quot represents the probability that the policyholder les n claims during the period Under this assumption what is the probability that a policyholder les more than one claim during the period A 004 B 016 020 080 C D H 9 E 096 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 5 coverage The proportions of the company s employees that choose coverages A B and C are 7 7 and E respectively Determine the probability that a randomly chosen employee will choose no supplementary coverage An insurance company determines that N the number of claims received in a week is a random variable with P N n TH 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 twoiweek period where n 2 0 The company also determines that the number of claims received in a given 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 25 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 B 010 C 020 025 040 080 D 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 00056h The error made by the more accurate instrument is normally distributed with mean 0 and standard deviation 00044h Assuming the two measure ments are independent random variables what is the probability that their average value is within 0005h of the height of the tower A 038 20 2 H B C D E 047 068 084 090 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 16720 006 008 21730 003 015 31765 002 049 66799 004 028 A randomly selected driver that the company insures has an accident Calculate the probability that the driver was age 16720 A B 013 016 019 O D E 023 040 An insurance company issues life insurance policies in three separate categories standard preferred and ultra preferred Of the companyAfs policyholders 50 are standard 40 are preferred and 10 are ultraipreferred Each standard policyholder has probability 0010 of dying in the next year each preferred policyholder has probability 0005 of dying in the next year and each ultraipreferred policyholder has probability 0001 of dying in the next year A policyholder dies in the next year What is the probability that the deceased policyholder was ultraipreferred Upon arrival at a hospital s emergency room patients are categorized according to their condition as critical serious or stable 1n the past year i 10 of the emergency room patients were critical ii 30 of the emergency room patients were serious iii the rest of the emergency room patients were stable iv 40 of the critical patients died vi 10 of the serious patients died and 3 2 4 vii 1 of the stable patients died Given that a patient survived what is the probability that the patient was categorized as serious upon arrival A B 006 C 029 030 039 064 D E A health study tracked a group of persons for ve years At the beginning of the study 20 were classi ed 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 veiyear study but only half as likely as heavy smokers A randomly selected participant from the study died over the veiyear period Calculate the probability that the participant was a heavy smoker A B 020 C 025 035 042 057 D E An actuary studied the likelihood that different types of drivers would be involved in at least one collision during any oneiyear period The results of the study are presented below Type of driver Percentage of all drivers Probability of at least one collision Teen 8 015 Young adult 16 008 Midlife 45 004 Senior 31 005 Total 100 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 B 006 C 016 019 022 025 D E 1 7 71 1n 27 where n 2 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 The number of injury claims per month is modeled by a random variable N with PN n A A U O V V mlmcmwwha 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 05 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 The probability that a randomly chosen male has a circulation problem is 025 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 problem7 given that he is a smoker E A A O 03 wlwleC lwleHgtlH A study of automobile accidents produced the following data Model year Proportion of all vehicles Probability of involvement in an accident 1997 016 005 1998 018 002 1999 020 003 Other 046 004 An automobile from one of the model years 19977 19987 and 1999 was involved in an accident Determine the probability that the model year of this automobile is 1997 A B 022 C 030 033 045 050 D E 28 A hospital receives 15 of its flu vaccine shipments from Company X and the remainder of its shipments from 2 3 3 3 Q3 0 H 3 other companies Each shipment contains a very large number of vaccine vials For Company X s shipments 10 of the vials are ineffective For every other company 2 of the vials are ineffective The hospital tests 30 randomly selected vials from a shipment and nds that one vial is ineffective What is the probability that this shipment came from Company X A B 010 C 014 037 063 086 D E The number of days that elapse between the beginning of a calendar year and the moment a highirisk driver is involved in an accident is exponentially distributed An insurance company expects that 30 of highirisk drivers will be involved in an accident during the rst 50 days of a calendar year What portion of highirisk drivers are expected to be involved in an accident during the rst 80 days of a calendar year A B 015 C 034 043 057 066 D E An actuary has discovered that policyholders are three times as likely to le two claims as to le four claims 1f the number of claims led has a Poisson distribution what is the variance of the number of claims led A 3 03 U E AAAAA Q 1 1 2 2 4 A company establishes a fund of 120 from which it wants to pay an amount C to any of its 20 employees who achieve a high performance level during the coming year Each employee has a 2 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 1 that the fund will be inadequate to cover all payments for high performance A 24 B 30 C 40 D 60 E 120 A large pool of adults earning their rst driver s license includes 50 lowerisk drivers 30 moderateirisk drivers and 20 highirisk 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 CA3 CT for adults earning their rst driver s license What is the probability that these 4 Will contain at least two more higherisk drivers than lowerisk drivers The loss due to a re in a commercial building is modeled by a random variable X With density function 000520 A x for 0 lt m lt 20 0 otherwise Given that a re loss exceeds 8 What is the probability that it exceeds 16 A A A A U Q 03 gt V V V V IlwleOolHRDlHlana H E The lifetime of a machine part has a continuous distribution on the interval 07 40 With probability density function f7 Where is proportional to 10 mfg Calculate the probability that the lifetime of the machine part is less than 6 A 004 B 015 c 047 053 094 D E The lifetime of a machine part has a continuous distribution on the interval 07 40 With probability density function f7 Where is proportional to 10 mfg What is the probability that the lifetime of the machine part is less than 5 A B 003 013 042 O D E 058 097 36 A group insurance policy covers the medical claims of the employees of a small company The value V of the 39 claims made in one year is described by V 100 000Y where Y is a random variable with density function k17y for0ltylt1 we 0 otherwise where k is a constant What is the conditional probability that V exceeds 40000 given that V exceeds 10000 A B 008 C 013 017 020 051 D E The lifetime of a printer costing 200 is exponentially distributed with mean 2 years The manufacturer agrees to pay a full refund to a buyer if the printer fails during the rst year following its purchase and a oneihalf refund if it fails during the second year If the manufacturer sells 100 printers how much should it expect to pay in refunds A 6321 B 7358 C 7869 D 10256 E 12642 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 39674 for m gt 1 0 otherwise Given that a randomly selected home is insured for at least 15 what is the probability that it is insured for less than 2 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 005 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 207year period 4 4 4 0 H 3 A B 006 C 019 038 062 092 D E An insurance policy pays for a random loss X subject to a deductible of C where 0 lt C lt 1 The loss amount is modeled as a continuous random variable with density function 296 for0ltxlt1 otherwise Given a random loss X the probability that the insurance payment is less than 05 is equal to 064 Calculate C A study is being conducted in which the health of two independent groups often policyholders is being monitored over a oneiyear period of time lndividual participants in the study drop out before the end of the study with probability 02 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 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 10000 and standard deviation 2000 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 9000 and standard deviation 2000 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 43 A company takes out an insurance policy to cover accidents that occur at its manufacturing plant 4 The probability that one or more accidents will occur during any given month is 7 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 B 001 012 023 O D E 029 041 An insurance policy pays 100 per day for up to 3 days of hospitalization and 50 per day for each day of hospii talization thereafter The number of days of hospitalization X is a discrete random variable with probability function 6 15 fork12345 0 otherwise Determine the expected payment for hospitalization under this policy A B 123 210 220 O D E 270 367 Let X be a continuous random variable with density function E for72 x 4 x 0 otherwise Calculate the expected value of X EE A EU chlwcan 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 rst two years of service the time to discovery of its failure is X maxT 2 Determine EX gt 1 23576 2 2 7 E g 4 55 03 MM D E 2 3 235 5 AAAA A Q A piece of equipment is being insured against early failure The time from purchase until failure of the equipment is exponentially distributed With mean 10 years The insurance Will pay an amount as if the equipment fails during the rst year and it Will pay 0596 if failure occurs during the second or third year If failure occurs after the rst three years no payment Will be made At What level must as be set if the expected payment made under this insurance is to be 1000 An insurance policy on an electrical device pays a bene t of 4000 if the device fails during the rst year The amount of the bene t decreases by 1000 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 04 What is the expected bene t under this policy An insurance policy pays an individual 100 per day for up to 3 days of hospitalization and 25 per day for each day of hospitalization thereafter The number of days of hospitalization X is a discrete random variable With probability function 67k T 0 otherwise for k 12345 Calculate the expected payment for hospitalization under this policy A B 85 C 163 168 213 255 D E 50 A company buys a policy to insure its revenue in the event of major snowstorms that shut down business The 53 policy pays nothing for the rst such snowstorm of the year and 10000 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 15 What is the expected amount paid to the company under this policy during a oneiyear period A manufacturer s annual losses follow a distribution with density function 250625 35 for m gt 06 f0 0 otherwise To cover its losses the manufacturer purchases an insurance policy with an annual deductible of 2 What is the mean of the manufacturer s annual losses not paid by the insurance policy A B 084 C 088 093 095 100 D E An insurance company sells a oneiyear automobile policy with a deductible of 2 The probability that the insured will incur a loss is 005 If there is a loss the probability of a loss of amount N is 7 for N 1 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 An insurance policy reimburses a loss up to a bene t limit of 10 distribution with density function 3 y The policyholder s loss Y follows a for y gt 1 f0 0 otherwise What is the expected value of the bene t paid under the insurance policy A 10 5 5 6 1 13 18 D19 20 An auto insurance company insures an automobile worth 15000 for one year under a policy with a 1000 deductible During the policy year there is a 004 chance of partial damage to the car and a 002 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 050035 for 0 lt x lt 15 NH 7 otherwise What is the expected claim payment A B 320 C 328 352 380 540 D E An insurance company s monthly claims are modeled by a continuous positive random variable X whose probability density function is proportional to 1 m 4 where 0 lt m lt 00 Determine the company s expected monthly claims E A U V CA3 leleH lH E An insurance policy is written to cover a loss X where X has a uniform distribution on 0 1000 At what level must a deductible be set in order for the expected payment to be 25 of what it would be with no deductible A B 250 C 375 500 625 750 D E An actuary determines that the claim size for a certain class of accidents is a random variable X with moment generating function 1 1 7 2500t439 Determine the standard deviation of the claim size for this class of accidents MXt 16 6 O A company insures homes in three cities J K and L Since suf cient distance separates the cities it is reasonable to assume that the losses occurring in these cities are independent functions for the loss distributions of the cities are The moment generating MJt 1 7 2t 3 MKt 1 7 2t 25 MLt 1 7 20quot An insurer s annual weather7related loss X is a random variable With density function 2520025 9635 for m gt 200 0 otherwise Calculate the difference between the 30th and 70th percentiles of X A recent study indicates that the annual cost of maintaining and repairing a car in a town in Ontario averages 200 With a variance of 260 If a tax of 20 is introduced on all items associated With the maintenance and repair of cars ie everything is made 20 more expensive What Will be the variance of the annual cost of maintaining and repairing a car A 208 B 260 61 62 c 270 D 312 E 374 An insurer s annual weatherirelated loss X is a random variable with density function 25 200 2395 for x gt 200 x n 0 otherwise Calculate the difference between the 25th and 75th percentiles of X A B 124 C 148 167 224 298 D E A random variable X has the cumulative distribution function f 0 if x lt 1 272 2 FWl if1 xlt2 1 Calculate the variance of X if71 m25 A A A A 9 9 E a lamewmwcsw A U V H 3 The warranty on a machine speci es that it will be replaced at failure or age 4 whichever occurs rst The machine s age at failure X has density function 1 r3 for0ltxlt5 x 0 otherw1se 1 Let Y be the age of the machine at the time of replacement Determine the variance of Y A 13 B 14 C 17 6 5 AA WU VV K 01 A probability distribution of the claim sizes for an auto insurance policy is given in the table below Claim Size Probability 20 015 30 010 40 005 50 020 60 010 70 010 80 030 What percentage of the claims are Within one standard deviation of the mean claim size The owner of an automobile insures it against damage by purchasing an insurance policy With a deductible of 250 1n the event that the automobile is damaged7 repair costs can be modeled by a uniform random variable on the interval 01500 Determine the standard deviation of the insurance payment in the event that the automobile is damaged A B 361 403 433 O D E 464 521 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 1 3 5 1sin7rx for i S m S i Which of the following represents the expected value of the accepted bid 5 5 A focos 39xdm 3 5 1 sin 7m4 dm H ml H miwgmm A 70 96 cos 7T L 1 sin 7m3 dx E i 1 Wmm A baseball team has scheduled its opening game for April 1 If it rains on April 17 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 1000 for each day7 up to 2 days7 that the opening game is postponed The insurance company determines that the number of consecutive days of rain beginning on April 1 is a Poisson random variable With mean 06 What is the standard deviation of the amount the insurance company Will have to pay A B 668 699 775 O D E 817 904 An insurance policy reimburses dental expense7 X7 up to a maximum bene t of 250 The probability density function for X is 0570004quot for m 2 0 0 otherwise Where c is a constant Calculate the median bene t for this policy A B 161 C 165 173 182 250 D E VVVVV 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 ve hours A B 007 029 038 O D E 042 057 An insurance company sells an auto insurance policy that covers losses incurred by a policyholder7 subject to a deductible of 100 Losses incurred follow an exponential distribution With mean 300 What is the 95th percentile of actual losses that exceed the deductible A B 600 700 800 O D E 900 1000 71 73 74 The time T that a manufacturing system is out of operation has cumulative distribution function 2 17ltggt fortgt2 0 otherwise ft The resulting cost to the company is Y T2 Determine the density function of Y for y gt 4 An investment account earns an annual interest rate R that follows a uniform distribution on the interval 004 008 The value of a 10000 initial investment in this account after one year is given by V 10 0005p Determine the cumulative distribution function Fu of V for values of u that satisfy 0 lt Fu lt 1 11 10 000510000 7 10408 425 B 255103300 7 004 11 7 10 408 l 10 833 7 10408 25 D 7 E 25 in m 7 004 An actuary models the lifetime of a device using the random variable Y 10X0398 where X is an exponential random variable with mean 1 year Determine the probability density function for y gt 0 of the random variable Y A 9 lt gt logo855W 8y7025710y03 C 8y7025701y1 25 01y12557012501y025 E 012501y039255 0391y125 Let T denote the time in minutes for a customer service representative to respond to 10 telephone inquiries T is uniformly distributed on the interval with endpoints 8 minutes and 12 minutes Let R denote the average rate in customers per minute at which the representative responds to inquiries Which of the following is the 10 10 density function of the random variable R on the interval E S r S A 132 77 B 3 i 5 c 3 7 51 l 13 2 E 2 The monthly pro t of Company I can be modeled by a continuous random variable with density function f Company H has a monthly pro t that is twice that of Company I Determine the probability density function of the monthly pro t of Company ll f 3 B f g M 2f lt2 D 2fw E WW A Claim amounts for wind damage to insured homes are independent random variables with common density function forxgt1 x4 x 0 otherwise where m is the amount of a claim in thousands Suppose 3 such claims will be made What is the expected value of the largest of the three claims A device runs until either of two components fails7 at which point the device stops running The joint density function of the lifetimes of the two components7 both measured in hours7 is my fxy for0ltmlt2and0ltylt2 What is the probability that the device fails during its rst hour of operation 78 A device runs until either of two components fails at which point the device stops running The joint density 7 8 8 9 O H function of the lifetimes of the two components both measured in hours is my 27 fxy for0ltxlt3and0ltylt3 Calculate the probability that the device fails during its rst hour of operation A B 004 041 044 O D E 059 096 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 fst where 0 lt s lt 1 and 0 lt t lt 1 What is the probability that the device fails during the rst half hour of operation 0505 ffst dsdt 0 A 39 5 fst ds dt A charity receives 2025 contributions Contributions are assumed to be independent and identically distributed with mean 3125 and standard deviation 250 Calculate the approximate 90t percentile for the distribution of the total contributions received A 6328000 B 6338000 C 6343000 D 6784000 E 6977000 Claims led under auto insurance policies follow a normal distribution with mean 19400 and standard deviation 5000 What is the probability that the average of 25 randomly selected claims exceeds 20000 A B 001 015 027 O D E 033 045 82 An insurance company issues 1250 vision care insurance policies The number of claims led by a policyholder under a vision care insurance policy during one year is a Poisson random variable with mean 2 Assume the numbers of claims led by distinct policyholders are independent of one another What is the approximate probability that there is a total of between 2450 and 2600 claims during a oneiyear period A 068 B 082 C 087 D 095 E 100 83 A company manufactures a brand of light bulb with a lifetime in months that is normally distributed with mean 3 and variance 1 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 09772 A B C D E 14 16 20 40 55 AAAAA VVVVV 84 Let X and Y be the number of hours that a randomly selected person watches movies and sporting events7 respectively7 during a threeimonth period The following information is known about X and Y EX 50 EY 20 VX 50 VY 30 COVXY 10 One hundred people are randomly selected and observed for these three months Let T be the total numi ber of hours that these one hundred people watch movies or sporting events during this threeimonth period Approximate the value of PT lt 7100 A 062 B 084 C 087 D 092 E 097 85 The total claim amount for a health insurance policy follows a distribution with density function e for m gt 0 24 8 8 8 7 00 The premium for the policy is set at 100 over the expected total claim amount If 100 policies are sold what is the approximate probability that the insurance company will have claims exceeding the premiums collected A city has just added 100 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 Each new recruit has a 04 probability of remaining with the police force until retirement ii Given that a new recruit reaches retirement with the police force the probability that she is not married at the time of retirement is 025 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 100 new hires and their husbands A B 060 C 067 075 093 099 D E In an analysis of health care 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 725 years to 25 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 025 years of the mean of the true ages A B 014 C 038 057 077 088 D E The waiting time for the rst claim from a good driver and the waiting time for the rst 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 rst claim from a good driver will be led within 3 years and the rst claim from a bad driver will be led within 2 years A 1 7 5723 7 5712 7 5776 i 18 7 776 B 185 C 1 7 5723 7 5712 7 5776 D 1 7 5723 7 5712 7 5713 An insurance company insures a large number of drivers 7 7237 712 i 776 E 1 35 65 185 The future lifetimes in months of two components of a machine have the following joint density function 507m7y for0ltmlt507ylt50 Jx7 y 7 0 otherwise What is the probability that both components are still functioning 20 months from now 6 i030 7 50 7 m 7 y dy dx 1257 000 0 0 l 6 30507x 125700020 20 6 305071711 C 7 gt12570002 JO 6 5f050f7x D 7 gt125700020 20 6 505071711 125 000 20 20 A B 50 7 m 7ydy dx 507x7ydydx 507m7ydydx E 50 7 m 7 y dy dx 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 Let X be the random variable representing the companyAfs losses under collision insurance7 and let Y represent the company s losses under liability insurance X and Y have joint density function 296 2 7 for0ltxlt1and0ltylt2 aw 0 otherwise What is the probability that the total loss is at least 1 A B 033 C 038 041 071 075 D E 92 Two insurers provide bids on an insurance policy to a large company The bids must be between 2000 and 2200 The company decides to accept the lower bid if the two bids differ by 20 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 2000 to 2200 Determine the probability that the company considers the two bids further A B 010 C 019 020 041 060 D E 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 10 One policy has a deductible of 1 and the other has a deductible of 2 The family experiences exactly one loss under each policy Calculate the probability that the total bene t paid to the family does not exceed 5 A B 013 025 030 O D E 032 042 Let T1 be the time between a car accident and reporting a claim to the insurance company Let T2 be the time between the report of the claim and payment of the claim The joint density function of T1 and T2 ft1t2 is constant over the region 0 lt 751 lt 6 0 lt 752 lt 6 751 752 lt 10 and zero otherwise Determine ET1 T2 the expected time between a car accident and payment of the claim A B 49 50 57 60 67 U AAAAA Di 0 VVVVV and Y are independent random variables with common moment generating function Mt etgZ Let X W X Y and Z Y 7 X Determine the joint moment generating function Mt1t2 of W and Z A 52z 2tg 5t1 t22 5t1t22 C A tour operator has a bus that can accommodate 20 tourists The operator knows that tourists may not show up so he sells 21 tickets The probability that an individual tourist will not show up is 002 independent of all other tourists Each ticket costs 50 and is nonerefundable if a tourist fails to show up If a tourist shows up and a seat is not available the tour operator has to pay 100 ticket cost 50 penalty to the tourist What is the expected revenue of the tour operator 99 A B 935 C 950 967 976 985 D E Let T1 and T2 represent the lifetimes in hours of two linked components in an electronic device The joint density function for T1 and T2 is uniform over the region de ned by 0 3 t1 3 t2 3 L where L is a positive constant Determine the expected value of the sum of the squares of T1 and T2 L2 E L2 Let X1 X2 X3 be a random sample from a discrete distribution with probability ifx0 mi 3 l 0 otherwise Determine the moment generating function M05 of Y X1X2X3 ifm1 19 t A 277 2775 B 1 25t 0 gay An insurance policy pays a total medical bene t consisting of two parts for each claim Let X represent the part of the bene t 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 10000 and the variance of the total bene t X Y is 17000 Due to increasing medical costs the company that issues the policy decides to increase X by a at amount of 100 per claim and to increase Y by 10 per claim Calculate the variance of the total bene t after these revisions have been made A 18200 B 18800 0 19300 100 10 H 102 D 19520 E 20670 A car dealership sells 0 1 or 2 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 HXQY HXLY HXLYD HX2Y HX2YD HX2Y What is the variance of X A B 047 C 058 083 142 258 D E The pro t for a new product is given by Z 3X 7 Y 7 5 X and Y are independent random variables with VarX 1 and VarY 2 What is the variance of Z A 03 U 1 E 6 AAAAA Q 1 5 7 1 1 A company has two electric generators The time until failure for each generator follows an exponential distrie bution with mean 10 The company will begin using the second generator immediately after the rst one fails What is the variance of the total time that the generators produce electricity A 10 B 20 C 50 D E 100 200 103 In a small metropolitan area annual losses due to storm re and theft are assumed to be independent 104 105 106 exponentially distributed random variables with respective means 10 15 and 24 Determine the probability that the maximum of these losses exceeds 3 A joint density function is given by km for0ltxlt1and0ltylt2 famp y 0 otherwise where k is a constant What is CouX Y A A 03 13gt V V l Ma wlw lH lH 0 Let X and Y be continuous random variables with joint density function 8 gm for0 x 1x y 2x f 967 y 0 otherwise Calculate the covariance of X and Y A B 004 C 025 067 080 124 D E Let X and Y denote the values of two stocks at the end of a veiyear period X is uniformly distributed on the interval 0 12 Given X x Y is uniformly distributed on the interval 0 Determine CouXY according to this model 107 108 109 110 E 24 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 EX 5 EX2 274 EY 7 EY2 514 and VarX Y 8 Let Cl X Y denote the size of the combined claims before the application of a 20 surcharge on the hospital portion of the claim and let Cg denote the size of the combined claims after the application of that surcharge Calculate CouC1 C2 A device containing two key components fails when and only when both components fail The lifetimes T1 and T2 of these components are independent with common density function e tt gt 0 The cost X of operating the device until failure is 2T1 T2 Which of the following is the density function of X for m gt 0 A 5 2 7 57quot 13 2w 7 54 x257 c T Six2 2 5713 3 A company offers earthquake insurance Annual premiums are modeled by an exponential random variable with mean 2 Annual claims are modeled by an exponential random variable with mean 1 Premiums and claims are independent Let X denote the ratio of claims to premiums What is the density function of X 1 2m 1 2 2m 12 A B Let X and Y be continuous random variables with joint density function r24xy for0ltxlt1and0ltylt17x ay i 0 otherwise Calculate P Y lt X X A 27 111 112 113 A A A U Q 03 V V V lugtcmgtaugtlgtaw Ilw E Once a re is reported to a re insurance company the company makes an initial estimate X of the amount it will pay to the claimant for the re loss When the claim is nally settled the company pays an amount Y to the claimant The company has determined that X and Y have the joint density function 2 Vb fmvy7x2x71y 1 1 formgt1ygt1 Given that the initial claim estimated by the company is 2 determine the probability that the nal settlement amount is between 1 and 3 E E A A U O V V oloowlwwlwsnlwunh 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 rst 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 fx y 22 y on the region where the density is positive Given that 10 of the employees buy the basic policy what is the probability that fewer than 5 buy the supplemental policy Two life insurance policies each with a death bene t of 10000 and a onetime 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 0025 the probability that only the husband will survive at least ten years is 001 and the probability that both of them will survive at least ten years is 096 What is the expected excess of premiums over claims given that the husband survives at least ten years A 350 B 385 c 397 114 115 116 D 870 E 897 A diagnostic test for the presence of a disease has two possible outcomes 1 for disease present and 0 for disease not present Let X denote the disease state of a patient7 and let Y denote the outcome of the diagnostic test The joint probability function of X and Y is given by PX 0 Y 0 0800 PX LY 0 0050 PX 0Y 1 0025 PX LY 1 0125 Calculate VarYlX 1 A 013 B 015 0 020 051 071 D 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 296 for0ltxlt1andxltyltx1 f w y 0 otherwise What is the conditional variance of Y given that X m E x2x 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 1 2 3 Annual number 0 012 006 005 002 of tornadoes 1 013 015 012 003 in county P 2 005 015 010 002 117 118 Calculate the conditional variance of the annual number of tornadoes in county Q7 given that there are no tornadoes in county P A B gt051 C 084 088 099 176 D 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 617xy forxgt07ygt07xylt1 otherwise Determine the probability that the portion of a claim representing damage to the house is less than 02 Let X and Y be continuous random variables with joint density function 15y foerSny aw 0 otherwise Let g be the marginal density function of Y Which of the following represents 9 15y for0ltylt1 A my 0 otherwise 15 2 Ty for m2 lt y lt x B 99 0 otherw1se 15 2 2y for 0 lt y lt 1 0 99 0 otherw1se 119 120 121 15y1 7y for m2 lt y lt m D my 0 otherwise 15y1 Ay for 0 lt y lt 1 E my 0 otherwise 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 car7 X7 has a marginal density function of 1 for 0 lt m lt 1 Given X x the size of the payment for damage to the other driver s car7 Y has conditional density of 1 for m lt y lt m 1 1f the policyholder is responsible for an accident7 what is the probability that the payment for damage to the other driver s car will be greater than 0500 A A A A U O US gt V V V V omuuMwleoolw A E SW An insurance policy is written to cover a loss X where X has density function 2962 for0 x 2 f w l 0 otherw1se The time in hours to process a claim of size as where 0 S m S 2 is uniformly distributed on the interval from m to 2x Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more A B 017 C 025 032 058 083 D E Let X represent the age of an insured automobile involved in an accident Let Y represent the length of time the owner has insured the automobile at the time of the accident X and Y have joint probability density function 1 a107xy2 for2 x 10and0 y 1 f w y otherwise 122 123 124 Calculate the expected age of an insured automobile involved in an accident A B M9 m2 ms m0 B64 A device contains two circuits The second circuit is a backup for the rst7 so the second is used only when the rst has failed The device fails when and only when the second circuit fails Let X and Y be the times at which the rst and second circuits fail7 respectively X and Y have joint probability density function f 6546211 for 0 lt m lt y lt 00 fltx7 y 0 otherwise What is the expected time at which the device fails A B 03 Q 050 067 083 150 D E You are given the following information about N7 the annual number of claims for a randomly selected insured HN HND HNgtD Let 5 denote the total annual claim amount for an insured When N 17 S is exponentially distributed with mean 5 When N gt 17 S is exponentially distributed with mean 8 Determine P4 lt S lt 8 A B OM Q 008 012 024 025 D E The joint probability density for X and Y is 2577 2y for m gt 0y gt 0 ml y 0 otherwise Calculate the variance of Y given that X gt 3 and Y gt 3 36 12 CH 126 127 A B gt025 C 050 100 325 350 D E The distribution of Y given X is uniform on the interval 0 X The marginal density of X is r 290 for 0 lt m lt 1 0 otherwise Determine the conditional density of X given Y y where positive A 03 V VVV 1 2 2 Q R D A AAA 1 y 1 17y E Under an insurance policy a maximum of ve claims may be led per year by a policyholder Let n p be the probability that a policyholder les n claims during a given year where n 0 1 23 45 An actuary makes the following observations 1 2 pn1f0r n 071727374 ii The difference between pH and pn1 is the same for n 0 1 2 3 4 iii Exactly 40 of policyholders le fewer than two claims during a given year Calculate the probability that a random policyholder will le more than three claims during a given year A B 014 016 027 O D E 029 033 Automobile losses reported to an insurance company are independent and uniformly distributed between 0 and 20000 The company covers each such loss subject to a deductible of 5000 Calculate the probability that the total payout on 200 reported losses is between 1000000 and 1200000 128 129 130 An insurance agent offers his clients auto insurance7 homeowners insurance and renters insurance The purchase of homeowners insurance and the purchase of renters insurance are mutually exclusive The pro le of the agent s clients is as follows i 17 of the clients have none of these three products ii 64 of the clients have auto insurance iii Twice as many of the clients have homeowners insurance as have renters insurance iv 35 of the clients have two of these three products v 11 of the clients have homeowners insurance7 but not auto insurance Calculate the percentage of the agent s clients that have both auto and renters insurance The cumulative distribution function for health care costs experienced by a policyholder is modeled by the function Fltx7 y 0 otherwise The policy has a deductible of 20 An insurer reimburses the policyholder for 100 of health care costs between 20 and 120 less the deductible Health care costs above 120 are reimbursed at 50 Let G be the cumulative distribution function of reimbursements given that the reimbursement is positive Calculate G115 The value of a piece of factory equipment after three years of use is 10005X where X is a random variable having moment generating function MXt 1 f tlt 1 or i 1727 2 ay 0 otherwise Calculate the expected value of this piece of equipment after three years of use A B 125 C 250 419 707 838 U E 131 132 133 134 Let N1 and N2 represent the numbers of claims submitted to a life insurance company in April and May respectively The joint probability function of N1 and N2 is 3 1 quot1 1 in 1 Z e 1175quot1quot2 for H1 123 and n2123 730117712 0 otherwise Calculate the expected number of claims that will be submitted to the company in May if exactly 2 claims were submitted in April 3 2 7 7 1 165 3 B E5 35 0475 D5271 E 2 5 2 A store has 80 modems in its inventory 30 coming from Source A and the remainder from Source B Of the modems from Source A 20 are defective Of the modems from Source B 8 are defective Calculate the probability that exactly two out of a random sample of ve modems from the storeAfs inventory are defective A man purchases a life insurance policy on his 40th birthday The policy will pay 5000 only if he dies before his 50th birthday and will pay 0 otherwise The length of lifetime in years of a male born the same year as the insured has the cumulative distribution function 0 for t S 0 1705 171 1 1 7 5 1000 fortgt0 Calculate the expected payment to the man under this policy A B 333 C 348 421 549 574 D E A mattress store sells only king queen and twinisize mattresses Sales records at the store indicate that one fourth as many queenisize mattresses are sold as king and twinsize mattresses combined Records also indicate that three times as many kingisize mattresses are sold as twinisize mattresses Calculate the probability that the next mattress sold is either king or queenisize 135 136 137 138 A B 012 c 015 080 085 095 D E VVVVV The number of workplace injuries N occurring in a factory on any given day is Poisson distributed with mean A The parameter A is a random variable that is determined by the level of activity in the factory and is uniformly distributed on the interval 03 Calculate VarN A B 2A C 075 150 225 D E A fair die is rolled repeatedly Let X be the number of rolls needed to obtain a 5 and Y the number of rolls needed to obtain a 6 Calculate EXlY 2 A B C D E 50 52 60 66 68 AAAAA VVVVV Let X and Y be identically distributed independent random variables such that the moment generating function of X Y is Mt 0095 2t 0245 034 024et 00952 for 700 lt t lt 00 Calculate PX S 0 A B 033 034 050 O D E 067 070 A machine consists of two components whose lifetimes have the joint density function 50 formgt0andmylt10 aw 0 otherwise The machine operates until both components fail Calculate the expected operational time of the machine A B C D E 17 25 33 50 67 AAAAA VVVVV 139 140 141 142 A driver and a passenger are in a car accident Each of them independently has probability 03 of being hospii talized When a hospitalization occurs the loss is uniformly distributed on 0 1 When two hospitalizations occur the losses are independent Calculate the expected number of people in the car who are hospitalized given that the total loss due to hospitalizations from the accident is less than 1 Each time a hurricane arrives a new home has a 04 probability of experiencing damage The occurrences of damage in different hurricanes are independent Calculate the mode of the number of hurricanes it takes for the home to experience damage from two hurricanes A 03 m4gtww E items are in a 67by75 array as shown ways to form a set of three distinct items such that no two of the selected items are in the same row or same column X is a continuous random variable with density function le for 7139031 otherwise Find E le A A A A O 03 Ilgt V V V V H com 90 O U 143 144 145 E 90 As part of the underwriting process for insurance each prospective policyholder is tested for diabetes Let X represent the number of tests completed when the rst person with diabetes pressure is found The expected value of X is 8 Calculate the probability that the fourth person tested is the rst one with high blood pressure In the casino game of roulette a wheel with 38 equally likely spots is spun and a ball is dropped at random into one of the 38 spots The 38 spots are numbers 1 to 36 along with 0 and 00 On a spin of the wheel a gambler can bet that the ball will drop into a speci ed spot If the ball does drop into that spot the gamble gets back the amount that he bet plus 36 time the amount that he bet If that spot does not turn up the gambler loses the amount bet A gambler can also bet that the outcome of the spin will be even 1f the ball drops into an even number spot from 2 to 36 the gambler gets back his bet plus an amount equal to the amount that he bet the bet is lost if the spot is 0 or 00 On every spin Gambler 1 always bets that the ball will drop in the spot with the number 1 and Gambler 2 always bets that the ball will drop into an even numbered spot Let X1 denote the net pro t of Gambler 1 after H spins and X2 denote the net pro t of Gambler 2 after the n spins Find EX2 7 X1 A 1 B 1 10 D 2 E 1 Fred Ned and Ted each have season tickets to the Toronto Rock Lacrosse Each one of them might or might not attend any particular game The probabilities describing their attendance for any particular game are Pat least one of them attends the game 95 Pat least two of them attends the game 80 Pall three of them attends the game 50 Their attendance pattern is also symmetric in the following way and PF NPF TPT N where F N and T denote the events that Fred Ned and Ted attended the game respectively For a particular game nd the probability that Fred and Ned attended gt O VVVV 15 30 45 60 75 US EU 146 147 148 X has pdf m for 0 lt m lt 1 Also7 i PX 0 1 ii PX 1 1 iii PX lt 0 PX gt 1 0 For What value of a is VarX maximized A 0 S a lt 1 B 1 S a lt 2 C 2 S a lt 3 D 3 S a lt 4 E a 2 4 The loss random variable X has the following pdf 96 7 g for 0 lt m S 1 x 7x1for1ltxlt2 0 otherwise l When a loss occurs an insurer pays the loss above a deductible of 5 up to a maximum insurance payment of 1 Find the insurer s expected payment When a loss occurs A 33 4 03 00 U0 39cn ox 6 80 1 E The loss random variable X has an exponential distribution With a mean of 0 gt 0 An insurance policy pays Y Where 1 fongo fw X l d EX forXgt0 E 5 gt O 1 571 1 254 1 7571 1 7 2571 1 i 5 1 03 U AAAA QD mm mm mm mm Di
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