SOA真题Course6ExamC(1)

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determine the bühlmann-straub empirical bayes estimate of the credibility factor z for
territory a.
(a) less than 0.4
(b) at least 0.4, but less than 0.5
(c) at least 0.5, but less than 0.6
(d) at least 0.6, but less than 0.7
(e) at least 0.7
exam c: fall 2005 -23- go on to next page
23. determine which of the following is a natural cubic spline passing through the three
points (0, y1 ), (1, y2 ), and (3, 6).
(a) ( )
( )
( )( ) ( )( ) ( )( )
3
2 3
3 7/6 , 0 1
2 1/6 1 11/6 1 11/24 1 , 1 3
x x x
f x
x x x x
= ⎧⎪⎨ − − ≤ <
+ − + − − − ≤ ≤ ⎪⎩
(b) ( )
( ) ( )( )
2 3
2 3
3 , 0 1
2 2 1 1/2 1 , 1 3
x x x x
f x
x x x
= ⎧⎪⎨ − − + ≤ <
+ − − − ≤ ≤ ⎪⎩
(c) ( )
( ) ( )
( )( ) ( ) ( )( )
2 3
2 3
3 1/2 1/2 , 0 1
2 1/2 1 1 1/8 1 , 1 3
x x x x
f x
x x x x
= ⎧⎪⎨ − − + ≤ <
− − + − − − ≤ ≤ ⎪⎩
(d) ( )
( ) ( ) ( )
( )( ) ( )( )
2 3
2 3
3 5/ 4 1/2 3/ 4 , 0 1
2 7/4 1 3/8 1 , 1 3
x x x x
f x
x x x
= ⎧⎪⎨ − − + ≤ <
+ − − − ≤ ≤ ⎪⎩
(e) ( )
( ) ( )
( )( ) ( )( )
3
2 3
3 3/2 1/2 , 0 1
2 3/ 2 1 1/ 4 1 , 1 3
x x x
f x
x x x
= ⎧⎪⎨ − + ≤ <
+ − − − ≤ ≤ ⎪⎩
exam c: fall 2005 -24- go on to next page
24. you are given:
(i) a cox proportional hazards model was used to study the survival times of
patients with a certain disease from the time of onset to death.
(ii) a single covariate z was used with z = 0 for a male patient and z = 1 for a female
patient.
(iii) a sample of five patients gave the following survival times (in months):
males: 10 18 25
females: 15 21
(iv) the parameter estimate is ˆβ= 0.27.
using the nelson-aalen estimate of the baseline cumulative hazard function, estimate the

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probability that a future female patient will survive more than 20 months from the time of
the onset of the disease.
(a) 0.33
(b) 0.36
(c) 0.40
(d) 0.43
(e) 0.50
exam c: fall 2005 -25- go on to next page

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(c) 0.06
(d) 0.09
(e) 0.12
exam c: fall 2005 -30- go on to next page
30. for a group of auto policyholders, you are given:
(i) the number of claims for each policyholder has a conditional poisson
distribution.
(ii) during year 1, the following data are observed for 8000 policyholders:
number of claims number of policyholders
0 5000
1 2100
2 750
3 100
4 50
5+ 0
a randomly selected policyholder had one claim in year 1.
determine the semiparametric empirical bayes estimate of the number of claims in
year 2 for the same policyholder.
(a) less than 0.15
(b) at least 0.15, but less than 0.30
(c) at least 0.30, but less than 0.45
(d) at least 0.45, but less than 0.60
(e) at least 0.60
exam c: fall 2005 -31- go on to next page
31. you are given:
(i) the following are observed claim amounts:
400 1000 1600 3000 5000 5400 6200
(ii) an exponential distribution with θ = 3300 is hypothesized for the data.
(iii) the goodness of fit is to be assessed by a p-p plot and a d(x) plot.
let (s, t) be the coordinates of the p-p plot for a claim amount of 3000.
determine (s−t)−d(3000).
(a) − 0.12
(b) − 0.08
(c) 0.00
(d) 0.08
(e) 0.12
exam c: fall 2005 -32- go on to next page

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