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历年真题选择Course6ExamC（三）——北美精算师考试SOA

来源：高顿网校 2014-07-22

　　请问有来报考今年10月份精算师的么？鼠标点击进来拜读吧——历年真题选择Course6ExamC（三）——北美精算师考试SOA
　　12. A smoothing spline is to be fit to the points (0, 3), (1, 2), and (3, 6).

　　The candidate function is:

　　( )

　　( ) ( )

　　( )( ) ( ) ( )( )

　　3

　　2 3

　　2.6 4/15 4/15 , 0 1

　　2.6 8/15 1 0.8 1 2/15 1 1 3

　　x x x

　　f x

　　x x x x

　　? = ?? ? + ≤ ≤

　　+ ? + ? ? ? ≤ ≤ ??

　　Determine the value of S, the squared norm smoothness criterion.

　　(A) Less than 2.35

　　(B) At least 2.35, but less than 2.50

　　(C) At least 2.50, but less than 2.65

　　(D) At least 2.65, but less than 2.80

　　(E) At least 2.80

　　Exam C: Fall 2005 -13- GO ON TO NEXT PAGE

　　13. You are given the following about a Cox proportional hazards model for mortality:

　　(i) There are two covariates: : z1 = 1 for smoker and 0 for non-smoker, and z2 =1 for

　　male and 0 for female.

　　(ii) The parameter estimates are 1

　　β? = 0.05 and 2

　　β? = 0.15 .

　　(iii) The covariance matrix of the parameter estimates, 1

　　?β

　　and 2

　　?β

　　, is:

　　? ??

　　?

　　? ??

　　?

　　0.0001 0.0003

　　0.0002 0.0001

　　Determine the upper limit of the 95% confidence interval for the relative risk of a female

　　non-smoker compared to a male smoker.

　　(A) Less than 0.6

　　(B) At least 0.6, but less than 0.8

　　(C) At least 0.8, but less than 1.0

　　(D) At least 1.0, but less than 1.2

　　(E) At least 1.2

　　Exam C: Fall 2005 -14- GO ON TO NEXT PAGE

　　14. You are given:

　　(i) Fifty claims have been observed from a lognormal distribution with unknown

　　parameters μ and σ .

　　(ii) The maximum likelihood estimates are ?? μ = 6.84 and σ?? = 1.49 .

　　(iii) The covariance matrix ofμ? and σ? is:

　　0 0444 0

　　0 0 0222

　　.

　　.

　　LN M

　　OQ P

　　(iv) The partial derivatives of the lognormal cumulative distribution function are:

　　F φ (z)

　　μ σ

　　? ?

　　=

　　?

　　and

　　?

　　?

　　=

　　F ?z× z

　　σ

　　φ

　　σ

　　b g

　　(v) An approximate 95% confidence interval for the probability that the next claim

　　will be less than or equal to 5000 is:

　　[PL, PH]

　　Determine PL.

　　(A) 0.73

　　(B) 0.76

　　(C) 0.79

　　(D) 0.82

　　(E) 0.85

　　Exam C: Fall 2005 -15- GO ON TO NEXT PAGE

　　15. For a particular policy, the conditional probability of the annual number of claims given

　　Θ = θ , and the probability distribution of Θ are as follows:

　　Number of Claims 0 1 2

　　Probability 2θ θ 1?3θ

　　θ 0.10 0.30

　　Probability 0.80 0.20

　　One claim was observed in Year 1.

　　Calculate the Bayesian estimate of the expected number of claims for Year 2.

　　(A) Less than 1.1

　　(B) At least 1.1, but less than 1.2

　　(C) At least 1.2, but less than 1.3

　　(D) At least 1.3, but less than 1.4

　　(E) At least 1.4

　　Exam C: Fall 2005 -16- GO ON TO NEXT PAGE

　　16. You simulate observations from a specific distribution F(x), such that the number of

　　simulations N is sufficiently large to be at least 95 percent confident of estimating

　　F(1500) correctly within 1 percent.

　　Let P represent the number of simulated values less than 1500.

　　Determine which of the following could be values of N and P.

　　(A) N = 2000 P = 1890

　　(B) N = 3000 P = 2500

　　(C) N = 3500 P = 3100

　　(D) N = 4000 P = 3630

　　(E) N = 4500 P = 4020

　　Exam C: Fall 2005 -17- GO ON TO NEXT PAGE

　　17. For a survival study, you are given:

　　(i) Deaths occurred at times s 1 2 9 y < y < … < y .

　　(ii) The Nelson-Aalen estimates of the cumulative hazard function at 3 y and 4 y are:

　　^ ^

　　H(y3)=0.4128 and H(y4)=0.5691

　　(iii) The estimated variances of the estimates in (ii) are:

　　^

　　Var[H(y3)] = 0.009565

　　∧

　　and ^

　　Var[H(y4 )] = 0.014448

　　∧

　　Determine the number of deaths at y4 .

　　(A) 2

　　(B) 3

　　(C) 4

　　(D) 5

　　(E) 6

　　Exam C: Fall 2005 -18- GO ON TO NEXT PAGE

　　18. A random sample of size n is drawn from a distribution with probability density function:

　　2 ( ) , 0 , 0

　　( )

　　f x x

　　x

　　θ

　　θ

　　θ

　　= < < ∞ >

　　+

　　Determine the asymptotic variance of the maximum likelihood estimator of θ .

　　(A) n

　　3θ 2

　　(B) 3 2

　　1

　　nθ

　　(C) 2

　　3

　　nθ

　　(D) 3 2

　　n

　　θ

　　(E) 3 2

　　1

　　θ

　　Exam C: Fall 2005 -19- GO ON TO NEXT PAGE
　　高顿网校之名人话语：一个人的风度和气质应使人近而敬之，而不要敬而远之，或近而轻之。——一凡

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