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(x|θ ) 2x, 0 x θ
θ
= < <
(ii) The prior distribution of θ has density function:
π (θ)=4θ 3, 0<θ <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) ˆ ( )
1 F x is the kernel density estimator of the distribution function using a uniform
kernel with bandwidth 1.
(iii) ˆ ( )
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 ˆ ( )
1 F x = ˆ ( )
2 F x for all x in the interval.
(A) 0 < x < 1
(B) 1 < x < 2
(C) 2 < x < 3
(D) 3 < x < 4
(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