32. You are given:
(i) In a portfolio of risks, each policyholder can have at most two claims per year.
(ii) For each year, the distribution of the number of claims is:
Number of Claims Probability
0 0.10
1 0.90− q
2 q
(iii) The prior density is:
2
( ) , 0.2 0.5
0.039
πq= q <q<
A randomly selected policyholder had two claims in Year 1 and two claims in Year 2.
For this insured, determine the Bayesian estimate of the expected number of claims in
Year 3.
(A) Less than 1.30
(B) At least 1.30, but less than 1.40
(C) At least 1.40, but less than 1.50
(D) At least 1.50, but less than 1.60
(E) At least 1.60
Exam C: Fall 2005 -33- GO ON TO NEXT PAGE
33. For 500 claims, you are given the following distribution:
Claim Size Number of Claims
[0, 500) 200
[500, 1,000) 110
[1,000, 2,000) x
[2,000, 5,000) y
[5,000, 10,000) ?
[10,000, 25,000) ?
[25,000, ∞) ?
You are also given the following values taken from the ogive:
F
500(1500) = 0.689
F
500(3500) = 0.839
Determine y.
(A) Less than 65
(B) At least 65, but less than 70
(C) At least 70, but less than 75
(D) At least 75, but less than 80
(E) At least 80
Exam C: Fall 2005 -34- GO ON TO NEXT PAGE
34. Which of statements (A), (B), (C), and (D) is false?
(A) The chi-square goodness-of-fit test works best when the expected number of
observations varies widely from interval to interval.
(B) For the Kolmogorov-Smirnov test, when the parameters of the distribution in the
null hypothesis are estimated from the data, the probability of rejecting the null
hypothesis decreases.
(C) For the Kolmogorov-Smirnov test, the critical value for right censored data should
be smaller than the critical value for uncensored data.
(D) The Anderson-Darling test does not work for grouped data.
(E) None of (A), (B), (C) or (D) is false.
Exam C: Fall 2005 -35- STOP
35. You are given:
(i) The number of claims follows a Poisson distribution.
(ii) Claim sizes follow a gamma distribution with parameters α (unknown) and
θ = 10,000 .
(iii) The number of claims and claim sizes are independent.
(iv) The full credibility standard has been selected so that actual aggregate losses will
be within 10% of expected aggregate losses 95% of the time.
Using limited fluctuation (classical) credibility, determine the expected number of claims
required for full credibility.
(A) Less than 400
(B) At least 400, but less than 450
(C) At least 450, but less than 500
(D) At least 500
(E) The expected number of claims required for full credibility cannot be determined
from the information given.
**END OF EXAMINATION**