SOA真题Course6ExamC(1)
question # answer question # answer
1 d 19 b
2 a 20 a
3 e 21 b
4 b 22 a
5 e 23 e
6 e 24 b and c
7 a 25 c
8 d 26 c
9 b 27 a
10 d and e 28 b
11 d 29 c
12 c 30 d
13 c 31 b
14 c 32 b
15 a 33 e
16 d 34 a
17 d 35 e
18 a
exam c: fall 2005 -1- go on to next page
**beginning of examination**
1. a portfolio of policies has produced the following claims:
100 100 100 200 300 300 300 400 500 600
determine the empirical estimate of h(300).
(a) less than 0.50
(b) at least 0.50, but less than 0.75
(c) at least 0.75, but less than 1.00
(d) at least 1.00, but less than 1.25
(e) at least 1.25
exam c: fall 2005 -2- go on to next page
2. you are given:
(i) the conditional distribution of the number of claims per policyholder is poisson with
mean λ .
(ii) the variable λ has a gamma distribution with parameters α and θ .
(iii) for policyholders with 1 claim in year 1, the credibility estimate for the number of
claims in year 2 is 0.15.
(iv) for policyholders with an average of 2 claims per year in year 1 and year 2, the
credibility estimate for the number of claims in year 3 is 0.20.
determine θ .
(a) less than 0.02
(b) at least 0.02, but less than 0.03
(c) at least 0.03, but less than 0.04
(d) at least 0.04, but less than 0.05
(e) at least 0.05
exam c: fall 2005 -3- go on to next page
3. a random sample of claims has been drawn from a burr distribution with known parameter
α = 1 and unknown parameters θ and γ . you are given:
(i) 75% of the claim amounts in the sample exceed 100.
(ii) 25% of the claim amounts in the sample exceed 500.
estimate θ by percentile matching.
(a) less than 190
(b) at least 190, but less than 200
(c) at least 200, but less than 210
(d) at least 210, but less than 220
(e) at least 220
exam c: fall 2005 -4- go on to next page
4. you are given:
(i) f ( x) is a cubic spline with knots (0, 0) and (2, 2).
(ii) f ′(0) = 1 and f ′′(2) = −24
determine f (1).
(a) 1
(b) 4
(c) 6
(d) 8
(e) 10
exam c: fall 2005 -5- go on to next page
5. for a portfolio of policies, you are given:
(i) there is no deductible and the policy limit varies by policy.
(ii) a sample of ten claims is:
350 350 500 500 500+ 1000 1000+ 1000+ 1200 1500
where the symbol + indicates that the loss exceeds the policy limit.
(iii) ^
s1(1250) is the product-limit estimate of s(1250).
(iv) ^
s2 (1250) is the maximum likelihood estimate of s(1250) under the assumption that
the losses follow an exponential distribution.
determine the absolute difference between ^
s1(1250) and ^
s2 (1250) .
(a) 0.00
(b) 0.03
(c) 0.05
(d) 0.08
(e) 0.09
exam c: fall 2005 -6- go on to next page
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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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