25. You are given:
(i) A random sample of losses from a Weibull distribution is:
595 700 789 799 1109
(ii) At the maximum likelihood estimates of θ and τ , Σln(f (xi )) = −33.05.
(iii) When τ = 2 , the maximum likelihood estimate of θ is 816.7.
(iv) You use the likelihood ratio test to test the hypothesis
H0:τ =2
H1:τ ≠2
Determine the result of the test.
(A) Do not reject 0 H at the 0.10 level of significance.
(B) Reject 0 H at the 0.10 level of significance, but not at the 0.05 level of
significance.
(C) Reject 0 H at the 0.05 level of significance, but not at the 0.025 level of
significance.
(D) Reject 0 H at the 0.025 level of significance, but not at the 0.01 level of
significance.
(E) Reject 0 H at the 0.01 level of significance.
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26. For each policyholder, losses X1,…, Xn , conditional on Θ, are independently and
identically distributed with mean,
( μ θ )=E(X jΘ=θ ), j=1,2,...,n
and variance,
v(θ)=Var(Xj Θ=θ), j = 1,2,...,n .
You are given:
(i) The Bühlmann credibility assigned for estimating X5 based on X1,…, X4 is
Z = 0.4.
(ii) The expected value of the process variance is known to be 8.
Calculate Cov(Xi, Xj), i≠ j.
(A) Less than −0.5
(B) At least −0.5 , but less than 0.5
(C) At least 0.5, but less than 1.5
(D) At least 1.5, but less than 2.5
(E) At least 2.5
Exam C: Fall 2005 -27- GO ON TO NEXT PAGE
27. Losses for a warranty product follow the lognormal distribution with underlying normal
mean and standard deviation of 5.6 and 0.75 respectively.
You use simulation to estimate claim payments for a number of contracts with different
deductibles.
The following are four uniform (0,1) random numbers:
0.6217 0.9941 0.8686 0.0485
Using these numbers and the inversion method, calculate the average payment per loss
for a contract with a deductible of 100.
(A) Less than 630
(B) At least 630, but less than 680
(C) At least 680, but less than 730
(D) At least 730, but less than 780
(E) At least 780
Exam C: Fall 2005 -28- GO ON TO NEXT PAGE
28. The random variable X has the exponential distribution with mean θ .
Calculate the e mean-squared error of X 2 as an estimator of θ 2 .
(A) 20θ 4
(B) 21θ 4
(C) 22θ 4
(D) 23θ 4
(E) 24θ 4
Exam C: Fall 2005 -29- GO ON TO NEXT PAGE
29. You are given the following data for the number of claims during a one-year period:
Number of Claims Number of Policies
0 157
1 66
2 19
3 4
4 2
5+ 0
Total 248
A geometric distribution is fitted to the data using maximum likelihood estimation.
Let P = probability of zero claims using the fitted geometric model.
A Poisson distribution is fitted to the data using the method of moments.
Let Q = probability of zero claims using the fitted Poisson model.
Calculate P Q − .
(A) 0.00
(B) 0.03
(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) − 0.07
(C) 0.00
(D) 0.07
(E) 0.12
Exam C: Fall 2005 -32- GO ON TO NEXT PAGE