以下是SOA北美精算师历年真题November2005ExamM（第五部分），请耐心看完哦。
　　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
　　高顿网校之淳淳教诲：每个人都被生命询问，而他只有用自己的生命才能回答此问题；只有以“负责”来答复生命。因此，“能够负责”是人类存在最重要的本质。——维克多·费兰克