下面的北美SOA精算师真题Course6ExamC往年第二部分，请相关SOA精算师的人士赶快消化掉。
　　6. The random variable X has survival function:
　　( )
　　( )
　　4
　　2 2 2
　　SX x
　　x
　　θ
　　θ
　　=
　　+
　　Two values of X are observed to be 2 and 4. One other value exceeds 4.
　　Calculate the maximum likelihood estimate of θ.
　　(A) Less than 4.0
　　(B) At least 4.0, but less than 4.5
　　(C) At least 4.5, but less than 5.0
　　(D) At least 5.0, but less than 5.5
　　(E) At least 5.5
　　Exam C: Fall 2005 -7- GO ON TO NEXT PAGE
　　7. For a portfolio of policies, you are given:
　　(i) The annual claim amount on a policy has probability density function:
　　2
　　f(xθ ) 2x, 0 x θ
　　θ
　　=
　　(ii) The prior distribution of θ has density function:
　　π (θ)=4θ 3, 0θ 1
　　(iii) A randomly selected policy had claim amount 0.1 in Year 1.
　　Determine the Bühlmann credibility estimate of the claim amount for the selected policy in
　　Year 2.
　　(A) 0.43
　　(B) 0.45
　　(C) 0.50
　　(D) 0.53
　　(E) 0.56
　　Exam C: Fall 2005 -8- GO ON TO NEXT PAGE
　　8. Total losses for a group of insured motorcyclists are simulated using the aggregate loss
　　model and the inversion method.
　　The number of claims has a Poisson distribution with λ = 4 . The amount of each claim has
　　an exponential distribution with mean 1000.
　　The number of claims is simulated using u = 0.13. The claim amounts are simulated using
　　u1 = 0.05 , u2 = 0.95 and u3 = 0.10 in that order, as needed.
　　Determine the total losses.
　　(A) 0
　　(B) 51
　　(C) 2996
　　(D) 3047
　　(E) 3152
　　Exam C: Fall 2005 -9- GO ON TO NEXT PAGE
　　9. You are given:
　　(i) The sample:
　　1 2 3 3 3 3 3 3 3 3
　　(ii) ? ( )
　　1 F x is the kernel density estimator of the distribution function using a uniform
　　kernel with bandwidth 1.
　　(iii) ? ( )
　　2 F x is the kernel density estimator of the distribution function using a triangular
　　kernel with bandwidth 1.
　　Determine which of the following intervals has ? ( )
　　1 F x = ? ( )
　　2 F x for all x in the interval.
　　(A) 0x1
　　(B) 1x2
　　(C) 2x3
　　(D) 3x4
　　(E) None of (A), (B), (C) or (D)
　　Exam C: Fall 2005 -10- GO ON TO NEXT PAGE
　　10. 1000 workers insured under a workers compensation policy were observed for one year. The
　　number of work days missed is given below:
　　Number of Days of Work
　　Missed
　　Number of Workers
　　0 818
　　1 153
　　2 25
　　3 or more 4
　　Total 1000
　　Total Number of Days Missed 230
　　The chi-square goodness-of-fit test is used to test the hypothesis that the number of work
　　days missed follows a Poisson distribution where:
　　(i) The Poisson parameter is estimated by the average number of work days missed.
　　(ii) Any interval in which the expected number is less than one is combined with the
　　previous interval.
　　Determine the results of the test.
　　(A) The hypothesis is not rejected at the 0.10 significance level.
　　(B) The hypothesis is rejected at the 0.10 significance level, but is not rejected at the 0.05
　　significance level.
　　(C) The hypothesis is rejected at the 0.05 significance level, but is not rejected at the
　　0.025 significance level.
　　(D) The hypothesis is rejected at the 0.025 significance level, but is not rejected at the
　　0.01 significance level.
　　(E) The hypothesis is rejected at the 0.01 significance level.
　　Exam C: Fall 2005 -11- GO ON TO NEXT PAGE
　　11. You are given the following data:
　　Year 1 Year 2
　　Total Losses 12,000 14,000
　　Number of Policyholders 25 30
　　The estimate of the variance of the hypothetical means is 254.
　　Determine the credibility factor for Year 3 using the nonparametric empirical Bayes method.
　　(A) Less than 0.73
　　(B) At least 0.73, but less than 0.78
　　(C) At least 0.78, but less than 0.83
　　(D) At least 0.83, but less than 0.88
　　(E) At least 0.88
　　Exam C: Fall 2005 -12- GO ON TO NEXT PAGE
　　高顿网校之名人感触：你是我人生中*10的主角，我却只能是你故事中的一晃而过得路人甲。