以下为高顿网校小编给全国考生准备的考前官方大纲要点——北美精算师2014年7月:SOA真题November2005ExamM（*9部分），希望真的可以帮到考友们。
　　Exam M Fall 2005
　　FINAL ANSWER KEY
　　Question # Answer Question # Answer
　　1 C 21 E
　　2 C 22 B
　　3 C 23 E
　　4 D 24 E
　　5 C 25 C
　　6 B 26 E
　　7 A 27 E
　　8 D 28 D
　　9 B 29 A
　　10 A 30 D
　　11 A 31 A
　　12 A 32 A
　　13 D 33 B
　　14 C 34 C
　　15 A 35 A
　　16 D 36 A
　　17 D 37 C
　　18 D 38 C
　　19 B 39 E
　　20 B 40 B
 
　　Exam M: Fall 2005 -1- GO ON TO NEXT PAGE
　　**BEGINNING OF EXAMINATION**
　　1. For a special whole life insurance on (x), you are given:
　　(i) Z is the present value random variable for this insurance.
　　(ii) Death benefits are paid at the moment of death.
　　(iii) ( ) 0.02, 0 xt t ? = ≥
　　(iv) 0.08 δ=
　　(v) 0.03, 0 t
　　tb e t = ≥
　　Calculate ( ) Var Z .
　　(A) 0.075
　　(B) 0.080
　　(C) 0.085
　　(D) 0.090
　　(E) 0.095
 
　　Exam M: Fall 2005 -2- GO ON TO NEXT PAGE
　　2. For a whole life insurance of 1 on (x), you are given:
　　(i) Benefits are payable at the moment of death.
　　(ii) Level premiums are payable at the beginning of each year.
　　(iii) Deaths are uniformly distributed over each year of age.
　　(iv) 0.10 i =
　　(v) 8 x a = ????
　　(vi) 10 6 x a + = ????
　　Calculate the 10th year terminal benefit reserve for this insurance.
　　(A) 0.18
　　(B) 0.25
　　(C) 0.26
　　(D) 0.27
　　(E) 0.30
 
　　Exam M: Fall 2005 -3- GO ON TO NEXT PAGE
　　3. A special whole life insurance of 100,000 payable at the moment of death of (x) includes a
　　double indemnity provision. This provision pays during the first ten years an additional
　　benefit of 100,000 at the moment of death for death by accidental means.
　　You are given:
　　(i) ? τ
　　x t t b gb g= ≥ 0 001 0 . ,
　　(ii) ?x t t 1 0 0002 0 b gb g= ≥ . , , where ?x
　　1 b g is the force of decrement due to death by
　　accidental means.
　　(iii) δ= 006 .
　　Calculate the single benefit premium for this insurance.
　　(A) 1640
　　(B) 1710
　　(C) 1790
　　(D) 1870
　　(E) 1970
 
　　Exam M: Fall 2005 -4- GO ON TO NEXT PAGE
　　4. Kevin and Kira are modeling the future lifetime of (60).
　　(i) Kevin uses a double decrement model:
　　x ( )
　　x l τ ( ) 1
　　x d ( ) 2
　　x d
　　60 1000 120 80
　　61 800 160 80
　　62 560 ? ?
　　(ii) Kira uses a non-homogeneous Markov model:
　　(a) The states are 0 (alive), 1 (death due to cause 1), 2 (death due to cause 2).
　　(b) 60 Q is the transition matrix from age 60 to 61; 61 Q is the transition matrix
　　from age 61 to 62.
　　(iii) The two models produce equal probabilities of decrement.
　　Calculate 61 Q .
　　(A)
　　1.00 0.12 0.08
　　0 1.00 0
　　0 0 1.00
　　? ?
　　? ?
　　? ?
　　? ?
　　? ?
　　(B)
　　0.80 0.12 0.08
　　0.56 0.16 0.08
　　0 0 1.00
　　? ?
　　? ?
　　? ?
　　? ?
　　? ?
　　(C)
　　0.76 0.16 0.08
　　0 1.00 0
　　0 0 1.00
　　? ?
　　? ?
　　? ?
　　? ?
　　? ?
　　(D)
　　0.70 0.20 0.10
　　0 1.00 0
　　0 0 1.00
　　? ?
　　? ?
　　? ?
　　? ?
　　? ?
　　(E)
　　0.60 0.28 0.12
　　0 1.00 0
　　0 0 1.00
　　? ?
　　? ?
　　? ?
　　? ?
　　? ?
　　Exam M: Fall 2005 -5- GO ON TO NEXT PAGE
　　5. A certain species of flower has three states: sustainable, endangered and extinct. Transitions
　　between states are modeled as a non-homogeneous Markov chain with transition matrices i Q
　　as follows:
　　1
　　Endangered Sustainable Extinct
　　Sustainable 0.85 0.15 0
　　0 0.7 0.3 Endangered
　　Extinct 0 0 1
　　Q
　　? ?
　　? ?
　　? ? = ? ?
　　? ?
　　? ?
　　? ?
　　2
　　0.9 0.1 0
　　0.1 0.7 0.2
　　0 0 1
　　Q
　　? ?
　　? ? =? ?
　　? ?
　　? ?
　　3
　　0.95 0.05 0
　　0.2 0.7 0.1
　　0 0 1
　　Q
　　[
　　15; ?
　　? ? =? ?
　　? ?
　　? ?
　　0.95 0.05 0
　　0.5 0.5 0 , 4,5,...
　　0 0 1
　　iQ i
　　? ?
　　? ? = = ? ?
　　? ?
　　? ?
　　Calculate the probability that a species endangered at the start of year 1 will ever become
　　extinct.
　　(A) 0.45
　　(B) 0.47
　　(C) 0.49
　　(D) 0.51
　　(E) 0.53
　　Exam M: Fall 2005 -6- GO ON TO NEXT PAGE
　　6. For a special 3-year term insurance:
　　(i) Insureds may be in one of three states at the beginning of each year: active, disabled,
　　or dead. All insureds are initially active. The annual transition probabilities are as
　　follows:
　　Active Disabled Dead
　　Active 0.8 0.1 0.1
　　Disabled 0.1 0.7 0.2
　　Dead 0.0 0.0 1.0
　　(ii) A 100,000 benefit is payable at the end of the year of death whether the insured was
　　active or disabled.
　　(iii) Premiums are paid at the beginning of each year when active. Insureds do not pay
　　any annual premiums when they are disabled.
　　(iv) d = 0.10
　　Calculate the level annual benefit premium for this insurance.
　　(A) 9,000
　　(B) 10,700
　　(C) 11,800
　　(D) 13,200
　　(E) 20,800
　　Exam M: Fall 2005 -7- GO ON TO NEXT PAGE
　　7. Customers arrive at a bank according to a Poisson process at the rate of 100 per hour. 20%
　　of them make only a deposit, 30% make only a withdrawal and the remaining 50% are there
　　only to complain. Deposit amounts are distributed with mean 8000 and standard deviation
　　1000. Withdrawal amounts have mean 5000 and standard deviation 2000.
　　The number of customers and their activities are mutually independent.
　　Using the normal approximation, calculate the probability that for an 8-hour day the total
　　withdrawals of the bank will exceed the total deposits.
　　(A) 0.27
　　(B) 0.30
　　(C) 0.33
　　(D) 0.36
　　(E) 0.39
　　Exam M: Fall 2005 -8- GO ON TO NEXT PAGE
　　8. A Mars probe has two batteries. Once a battery is activated, its future lifetime is exponential
　　with mean 1 year.
　　The first battery is activated when the probe lands on Mars. The second battery is activated
　　when the first fails.
　　Battery lifetimes after activation are independent.
　　The probe transmits data until both batteries have failed.
　　Calculate the probability that the probe is transmitting data three years after landing.
　　(A) 0.05
　　(B) 0.10
　　(C) 0.15
　　(D) 0.20
　　(E) 0.25
　　Exam M: Fall 2005 -9- GO ON TO NEXT PAGE
　　9. For a special fully discrete 30-payment whole life insurance on (45), you are given:
　　(i) The death benefit of 1000 is payable at the end of the year of death.
　　(ii) The benefit premium for this insurance is equal to 45 1000P for the first 15 years
　　followed by an increased level annual premium of π for the remaining 15 years.
　　(iii) Mortality follows the Illustrative Life Table.
　　(iv) 0.06 i =
　　Calculate π.
　　(A) 16.8
　　(B) 17.3
　　(C) 17.8
　　(D) 18.3
　　(E) 18.8
　　Exam M: Fall 2005 -10- GO ON TO NEXT PAGE
　　10. For a special fully discrete 2-year endowment insurance on (x):
　　(i) The pure endowment is 2000.
　　(ii) The death benefit for year k is ( ) 1000k plus the benefit reserve at the end of year k,
　　1, 2 k = .
　　(iii) π is the level annual benefit premium.
　　(iv) i = 0.08
　　(v) 1 0.9, 1, 2 x k p k + ? = =
　　Calculate π.
　　(A) 1027
　　(B) 1047
　　(C) 1067
　　(D) 1087
　　(E) 1107
　　Exam M: Fall 2005 -11- GO ON TO NEXT PAGE
　　11. For a group of 250 individuals age x, you are given:
　　(i) The future lifetimes are independent.
　　(ii) Each individual is paid 500 at the beginning of each year, if living.
　　(iii) 0.369131 x A =
　　(iv) 2 0.1774113 x A =
　　(v) 0.06 i =
　　Using the normal approximation, calculate the size of the fund needed at inception in order to
　　be 90% certain of having enough money to pay the life annuities.
　　(A) 1.43 million
　　(B) 1.53 million
　　(C) 1.63 million
　　(D) 1.73 million
　　(E) 1.83 million
　　Exam M: Fall 2005 -12- GO ON TO NEXT
　　PAGE
　　12. For a double decrement table, you are given:
　　Age ( )
　　x l τ ( ) 1
　　x d ( ) 2
　　x d
　　40 1000 60 55
　　41 ? ? 70
　　42 750 ? ?
　　Each decrement is uniformly distributed over each year of age in the double decrement table.
　　Calculate ( ) 1
　　41 q′ .
　　(A) 0.077
　　(B) 0.078
　　(C) 0.079
　　(D) 0.080
　　(E) 0.081
　　Exam M: Fall 2005 -13- GO ON TO NEXT PAGE
　　13. The actuarial department for the SharpPoint Corporation models the lifetime of pencil
　　sharpeners from purchase using a generalized DeMoivre model with ( ) ( ) 1 / s x x α ω = ? , for
　　0 α> and 0 x ω ≤ ≤ .
　　A senior actuary examining mortality tables for pencil sharpeners has determined that the
　　original value of α must change. You are given:
　　(i) The new complete expectation of life at purchase is half what it was previously.
　　(ii) The new force of mortality for pencil sharpeners is 2.25 times the previous force of
　　mortality for all durations.
　　(iii) ω remains the same.
　　Calculate the original value of α.
　　(A) 1
　　(B) 2
　　(C) 3
　　(D) 4
　　(E) 5
　　Exam M: Fall 2005 -14- GO ON TO NEXT PAGE
　　14. You are given:
　　(i) T is the future lifetime random variable.
　　(ii) ( ) t ? ? = , 0 t ≥
　　(iii) [ ] Var 100 T = .
　　Calculate [ ] E 10 T ∧ .
　　(A) 2.6
　　(B) 5.4
　　(C) 6.3
　　(D) 9.5
　　(E) 10.0
　　Exam M: Fall 2005 -15- GO ON TO NEXT PAGE
　　15. For a fully discrete 15-payment whole life insurance of 100,000 on (x), you are given:
　　(i) The expense-loaded level annual premium using the equivalence principle is 4669.95.
　　(ii) 100,000 51,481.97 x A =
　　(iii) :15 11.35 x a = ????
　　(iv) 0.02913 d =
　　(v) Expenses are incurred at the beginning of the year.
　　(vi) Percent of premium expenses are 10% in the first year and 2% thereafter.
　　(vii) Per policy expenses are K in the first year and 5 in each year thereafter until death.
　　Calculate K.
　　(A) 10.0
　　(B) 16.5
　　(C) 23.0
　　(D) 29.5
　　(E) 36.5
　　Exam M: Fall 2005 -16- GO ON TO NEXT PAGE
　　16. For the future lifetimes of (x) and (y):
　　(i) With probability 0.4, ( ) ( ) T x T y = (i.e., deaths occur simultaneously).
　　(ii) With probability 0.6, the joint density function is
　　( ) ( ) , ( , ) 0.0005 T x T y f ts= , 0 40 t < < , 0 50 s < <
　　Calculate ( ) ( ) Prob T x T y < ? ? ? ?.
　　(A) 0.30
　　(B) 0.32
　　(C) 0.34
　　(D) 0.36
　　(E) 0.38
　　Exam M: Fall 2005 -17- GO ON TO NEXT PAGE
　　17. The length of time, in years, that a person will remember an actuarial statistic is modeled by
　　an exponential distribution with mean 1
　　Y . In a certain population, Y has a gamma
　　distribution with 2 α θ = = .
　　Calculate the probability that a person drawn at random from this population will remember
　　an actuarial statistic less than 1
　　2 year.
　　(A) 0.125
　　(B) 0.250
　　(C) 0.500
　　(D) 0.750
　　(E) 0.875
　　Exam M: Fall 2005 -18- GO ON TO NEXT PAGE
　　18. In a CCRC, residents start each month in one of the following three states: Independent
　　Living (State #1), Temporarily in a Health Center (State #2) or Permanently in a Health
　　Center (State #3). Transitions between states occur at the end of the month.
　　If a resident receives physical therapy, the number of sessions that the resident receives in a
　　month has a geometric distribution with a mean which depends on the state in which the
　　resident begins the month. The numbers of sessions received are independent. The number
　　in each state at the beginning of a given month, the probability of needing physical therapy in
　　the month, and the mean number of sessions received for residents receiving therapy are
　　displayed in the following table:
　　State # Number in
　　state
　　Probability of
　　needing therapy
　　Mean number
　　of visits
　　1 400 0.2 2
　　2 300 0.5 15
　　3 200 0.3 9
　　Using the normal approximation for the aggregate distribution, calculate the probability that
　　more tha
　　n 3000 physical therapy sessions will be required for the given month.
　　(A) 0.21
　　(B) 0.27
　　(C) 0.34
　　(D) 0.42
　　(E) 0.50
　　Exam M: Fall 2005 -19- GO ON TO NEXT PAGE
　　19. In a given week, the number of projects that require you to work overtime has a geometric
　　distribution with 2 β= . For each project, the distribution of the number of overtime hours in
　　the week is the following:
　　x ( ) f x
　　5 0.2
　　10 0.3
　　20 0.5
　　The number of projects and number of overtime hours are independent. You will get paid for
　　overtime hours in excess of 15 hours in the week.
　　Calculate the expected number of overtime hours for which you will get paid in the week.
　　(A) 18.5
　　(B) 18.8
　　(C) 22.1
　　(D) 26.2
　　(E) 28.0
　　Exam M: Fall 2005 -20- GO ON TO NEXT PAGE
　　20. For a group of lives age x, you are given:
　　(i) Each member of the group has a constant force of mortality that is drawn from the
　　uniform distribution on [ ] 0.01, 0.02 .
　　(ii) 0.01 δ=
　　For a member selected at random from this group, calculate the actuarial present value of a
　　continuous lifetime annuity of 1 per year.
　　(A) 40.0
　　(B) 40.5
　　(C) 41.1
　　(D) 41.7
　　(E) 42.3
　　Exam M: Fall 2005 -21- GO ON TO NEXT PAGE
　　21. For a population whose mortality follows DeMoivre’s law, you are given:
　　(i) 40:40 60:60 3 e e = ?? ??
　　(ii) 20:20 60:60 e ke = ?? ??
　　Calculate k.
　　(A) 3.0
　　(B) 3.5
　　(C) 4.0
　　(D) 4.5
　　(E) 5.0
　　Exam M: Fall 2005 -22- GO ON TO NEXT PAGE
　　22. For an insurance on (x) and (y):
　　(i) Upon the first death, the survivor receives the single benefit premium for a whole life
　　insurance of 10,000 payable at the moment of death of the survivor.
　　(ii) ( ) ( ) 0.06 x y t t ? ? = = while both are alive.
　　(iii) ( ) 0.12 x y t ? =
　　(iv) After the first death, ( ) 0.10 t ? = for the survivor.
　　(v) 0.04 δ=
　　Calculate the actuarial present value of this insurance on (x) and (y).
　　(A) 4500
　　(B) 5400
　　(C) 6000
　　(D) 7100
　　(E) 7500
　　Exam M: Fall 2005 -23- GO ON TO NEXT PAGE
　　23. Kevin and Kira are in a history competition:
　　(i) In each round, every child still in the contest faces one question. A child is out as
　　soon as he or she misses one question. The contest will last at least 5 rounds.
　　(ii) For each question, Kevin’s probability and Kira’s probability of answering that
　　question correctly are each 0.8; their answers are independent.
　　Calculate the conditional probability that both Kevin and Kira are out by the start of round
　　five, given that at least one of them participates in round 3.
　　(A) 0.13
　　(B) 0.16
　　(C) 0.19
　　(D) 0.22
　　(E) 0.25
　　Exam M: Fall 2005 -24- GO ON TO NEXT PAGE
　　24. For a special increasing whole life annuity-due on (40), you are given:
　　(i) Y is the present-value random variable.
　　(ii) Payments are made once every 30 years, beginning immediately.
　　(iii) The payment in year 1 is 10, and payments increase by 10 every 30 years.
　　(iv) Mortality follows DeMoivre’s law, with 110 ω= .
　　(v) 0.04 i =
　　Calculate ( ) Var Y .
　　(A) 10.5
　　(B) 11.0
　　(C) 11.5
　　(D) 12.0
　　(E) 12.5
　　Exam M: Fall 2005 -25- GO ON TO NEXT PAGE
　　25. For a special 3-year term insurance on ( ) x , you are given:
　　(i) Z is the present-value random variable for this insurance.
　　(ii) q k x k + = + 002 1 . ( ), k = 0, 1, 2
　　(iii) The following benefits are payable at the end of the year of death:
　　k bk+1
　　0 300
　　1 350
　　2 400
　　(iv) i = 006 .
　　Calculate Var Z b g .
　　(A) 9,600
　　(B) 10,000
　　(C) 10,400
　　(D) 10,800
　　(E) 11,200
　　Exam M: Fall 2005 -26- GO ON TO NEXT PAGE
　　26. For an insurance:
　　(i) Losses have density function
　　( ) 0.02 0 10
　　0 elsewhere X
　　x x
　　f x
　　< < ?
　　= ??
　　(ii) The insurance has an ordinary deductible of 4 per loss.
　　(iii) P Y is the claim payment per payment random variable.
　　Calculate E P Y ? ?
　　? ? .
　　(A) 2.9
　　(B) 3.0
　　(C) 3.2
　　(D) 3.3
　　(E) 3.4
　　Exam M: Fall 2005 -27- GO ON TO NEXT PAGE
　　27. An actuary has created a compound claims frequency model with the following properties:
　　(i) The primary distribution is the negative binomial with probability generating function
　　( ) ( ) 2 1 3 1 P z z ? = ? ? ? ? ? ? .
　　(ii) The secondary distribution is the Poisson with probability generating function
　　( ) ( ) 1 z P z eλ ? = .
　　(iii) The probability of no claims equals 0.067.
　　Calculate λ.
　　(A) 0.1
　　(B) 0.4
　　(C) 1.6
　　(D) 2.7
　　(E) 3.1
　　Exam M: Fall 2005 -28- GO ON TO NEXT PAGE
　　28. In 2005 a risk has a two-parameter Pareto distribution with 2 α= and 3000 θ= . In 2006
　　losses inflate by 20%.
　　An insurance on the risk has a deductible of 600 in each year. i P , the premium in year i,
　　equals 1.2 times the expected claims.
　　The risk is reinsured with a deductible that stays the same in each year. i R , the reinsurance
　　premium in year i, equals 1.1 times the expected reinsured claims.
　　2005
　　2005 0.55 R
　　P =
　　Calculate 2006
　　2006
　　R
　　P .
　　(A) 0.46
　　(B) 0.52
　　(C) 0.55
　　(D) 0.58
　　(E) 0.66
　　Exam M: Fall 2005 -29- GO ON TO NEXT PAGE
　　29. For a fully discrete whole life insurance of 1000 on (60), you are given:
　　(i) The expenses, payable at the beginning of the year, are:
　　Expense Type First Year Renewal Years
　　% of Premium 20% 6%
　　Per Policy 8 2
　　(ii) The level expense-loaded premium is 41.20.
　　(iii) i = 0.05
　　Calculate the value of the expense augmented loss variable, 0 e L , if the insured dies in the
　　third policy year.
　　(A) 770
　　(B) 790
　　(C) 810
　　(D) 830
　　(E) 850
　　Exam M: Fall 2005 -30- GO ON TO NEXT PAGE
　　30. For a fully discrete whole life insurance of 1000 on (45), you are given:
　　t 45 1000tV 45 t q +
　　22 235 0.015
　　23 255 0.020
　　24 272 0.025
　　Calculate 25 45 1000 V .
　　(A) 279
　　(B) 282
　　(C) 284
　　(D) 286
　　(E) 288
　　Exam M: Fall 2005 -31- GO ON TO NEXT PAGE
　　31. The graph of a piecewise linear survival function, ( ) s x , consists of 3 line segments with
　　endpoints (0, 1), (25, 0.50), (75, 0.40), (100, 0).
　　Calculate 15 20 55
　　55 35
　　q
　　q
　　.
　　(A) 0.69
　　(B) 0.71
　　(C) 0.73
　　(D) 0.75
　　(E) 0.77
　　Exam M: Fall 2005 -32- GO ON TO NEXT PAGE
　　32. For a group of lives aged 30, containing an equal number of smokers and non-smokers, you
　　are given:
　　(i) For non-smokers, ( ) 0.08 n x ? = , 30 x ≥
　　(ii) For smokers, ( ) 0.16, s x ? = 30 x ≥
　　Calculate 80 q for a life randomly selected from those surviving to age 80.
　　(A) 0.078
　　(B) 0.086
　　(C) 0.095
　　(D) 0.104
　　(E) 0.112
　　Exam M: Fall 2005 -33- GO ON TO NEXT PAGE
　　33. For a 3-year fully discrete term insurance of 1000 on (40), subject to a double decrement
　　model:
　　(i)
　　x ( )
　　x l τ ( ) 1
　　x d ( ) 2
　　x d
　　40 2000 20 60
　　41 ? 30 50
　　42 ? 40 ?
　　(ii) Decrement 1 is death. Decrement 2 is withdrawal.
　　(iii) There are no withdrawal benefits.
　　(iv) 0.05 i =
　　Calculate the level annual benefit premium for this insurance.
　　(A) 14.3
　　(B) 14.7
　　(C) 15.1
　　(D) 15.5
　　(E) 15.7
　　Exam M: Fall 2005 -34- GO ON TO NEXT PAGE
　　34. Each life within a group medical expense policy has loss amounts which follow a compound
　　Poisson process with 0.16 λ= . Given a loss, the probability that it is for Dis
　　ease 1 is 1
　　16 .
　　Loss amount distributions have the following parameters:
　　Mean per loss
　　Standard
　　Deviation per loss
　　Disease 1 5 50
　　Other diseases 10 20
　　Premiums for a group of 100 independent lives are set at a level such that the probability
　　(using the normal approximation to the distribution for aggregate losses) that aggregate
　　losses for the group will exceed aggregate premiums for the group is 0.24.
　　A vaccine which will eliminate Disease 1 and costs 0.15 per person has been discovered.
　　Define:
　　A = the aggregate premium assuming that no one obtains the vaccine, and
　　B = the aggregate premium assuming that everyone obtains the vaccine and the cost of the
　　vaccine is a covered loss.
　　Calculate A/B.
　　(A) 0.94
　　(B) 0.97
　　(C) 1.00
　　(D) 1.03
　　(E) 1.06
　　Exam M: Fall 2005 -35- GO ON TO NEXT PAGE
　　35. An actuary for a medical device manufacturer initially models the failure time for a particular
　　device with an exponential distribution with mean 4 years.
　　This distribution is replaced with a spliced model whose density function:
　　(i) is uniform over [0, 3]
　　(ii) is proportional to the initial modeled density function after 3 years
　　(iii) is continuous
　　Calculate the probability of failure in the first 3 years under the revised distribution.
　　(A) 0.43
　　(B) 0.45
　　(C) 0.47
　　(D) 0.49
　　(E) 0.51
　　Exam M: Fall 2005 -36- GO ON TO NEXT PAGE
　　36. For a fully continuous whole life insurance of 1 on (30), you are given:
　　(i) The force of mortality is 0.05 in the first 10 years and 0.08 thereafter.
　　(ii) 0.08 δ=
　　Calculate the benefit reserve at time 10 for this insurance.
　　(A) 0.144
　　(B) 0.155
　　(C) 0.166
　　(D) 0.177
　　(E) 0.188
　　Exam M: Fall 2005 -37- GO ON TO NEXT PAGE
　　37. For a 10-payment, 20-year term insurance of 100,000 on Pat:
　　(i) Death benefits are payable at the moment of death.
　　(ii) Contract premiums of 1600 are payable annually at the beginning of each year for 10
　　years.
　　(iii) i = 0.05
　　(iv) L is the loss random variable at the time of issue.
　　Calculate the minimum value of L as a function of the time of death of Pat.
　　(A) ? 21,000
　　(B) ? 17,000
　　(C) ? 13,000
　　(D) ? 12,400
　　(E) ? 12,000
　　Exam M: Fall 2005 -38- GO ON TO NEXT PAGE
　　38. For an insurance:
　　(i) The number of losses per year has a Poisson distribution with 10 λ= .
　　(ii) Loss amounts are uniformly distributed on (0, 10).
　　(iii) Loss amounts and the number of losses are mutually independent.
　　(iv) There is an ordinary deductible of 4 per loss.
　　Calculate the variance of aggregate payments in a year.
　　(A) 36
　　(B) 48
　　(C) 72
　　(D) 96
　　(E) 120
　　Exam M: Fall 2005 -39- GO ON TO NEXT PAGE
　　39. For an insurance portfolio:
　　(i) The number of claims has the probability distribution
　　n n p
　　0 0.1
　　1 0.4
　　2 0.3
　　3 0.2
　　(ii) Each claim amount has a Poisson distribution with mean 3; and
　　(iii) The number of claims and claim amounts are mutually independent.
　　Calculate the variance of aggregate claims.
　　(A) 4.8
　　(B) 6.4
　　(C) 8.0
　　(D) 10.2
　　(E) 12.4
　　Exam M: Fall 2005 -40- STOP
　　40. Lucky Tom deposits the coins he finds on the way to work according to a Poisson process
　　with a mean of 22 deposits per month.
　　5% of the time, Tom deposits coins worth a total of 10.
　　15% of the time, Tom deposits coins worth a total of 5.
　　80% of the time, Tom deposits coins wort
　　高顿网校之心灵鸡汤：一个有事业追求的人，可以把“梦”做得高些。虽然开始时是梦想，但只要不停地做，不轻易放弃，梦想能成真。