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19. For a portfolio of independent risks, the number of claims for each risk in a year follows
a Poisson distribution with means given in the following table:
Class
Mean Number of
Claims per Risk Number of Risks
1 1 900
2 10 90
3 20 10
You observe x claims in Year 1 for a randomly selected risk.
The Bühlmann credibility estimate of the number of claims for the same risk in Year 2 is
11.983.
Determine x.
(A) 13
(B) 14
(C) 15
(D) 16
(E) 17
Exam C: Fall 2005 -20- GO ON TO NEXT PAGE
20. A survival study gave (0.283, 1.267) as the symmetric linear 95% confidence interval
for H(5).
Using the delta method, determine the symmetric linear 95% confidence interval for S(5).
(A) (0.23, 0.69)
(B) (0.26, 0.72)
(C) (0.28, 0.75)
(D) (0.31, 0.73)
(E) (0.32, 0.80)
Exam C: Fall 2005 -21- GO ON TO NEXT PAGE
21. You are given:
(i) Losses on a certain warranty product in Year i follow a lognormal distribution
with parameters i
μ and i
σ .
(ii) i
σ =σ , for i = 1, 2, 3,…
(iii) The parameters i
μ vary in such a way that there is an annual inflation rate of 10%
for losses.
(iv) The following is a sample of seven losses:
Year 1: 20 40 50
Year 2: 30 40 90 120
Using trended losses, determine the method of moments estimate of μ3 .
(A) 3.87
(B) 4.00
(C) 30.00
(D) 55.71
(E) 63.01
Exam C: Fall 2005 -22- GO ON TO NEXT PAGE
22. You are given:
(i) A region is comprised of three territories. Claims experience for Year 1 is as
follows:
Territory Number of Insureds Number of Claims
A 10 4
B 20 5
C 30 3
(ii) The number of claims for each insured each year has a Poisson distribution.
(iii) Each insured in a territory has the same expected claim frequency.
(iv) The number of insureds is constant over time for each territory.
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
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