CREDIBILITY - PROBLEM SET 2
Bayesian Analysis - Discrete Prior
Questions 1 and 2 relate to the following situation. Two bowls each contain 10 similarly shaped
balls. Bowl 1 contains 5 red and 5 white balls (equally likely to be chosen). Bowl 2 contains 2
14.
For a group of insureds, you are given: (i) The amount of a claim is uniformly distributed but will not exceed a certain unknown limit . The prior distribution of is =
(ii) (iii)
bg
500 , > 500 . 2
Two independent claims of 400 and 600 are observed.
D
CREDIBILITY - PROBLEM SET 6
Buhlmann Credibility
1. Assume that the number of claims each year for an individual insured has a Poisson distribution. Assume also that the expected annual claim frequencies (the Poisson parameters A) of the members of the po
ACT466H1S - TEST 3 - APRIL 10, 2007 Write name and student number on each page. Write your solution for each question in the space provided.
4
1. (a) The prior distribution of - is a gamma distribution with parameters oe $ and ) oe #. The conditional dist
ACT466H1S - TEST 3 - APRIL 12, 2007 Write name and student number on each page. Write your solution for each question in the space provided.
5
5
1. W has a compound distribution. The frequency distribution R has probability function 8 ! " # $ T R 8 % $ #
ACT466H1S - TEST 2 - MARCH 22, 2007 Write name and student number on each page. Write your solution for each question in the space provided.
1. The parameter : has a prior distribution with pdf 1: oe #: for ! : " . The model distribution \ is a discrete n
19. The distribution of \ in three consecutive periods has the following characteristics: I\" " Z +<\" " I\# # Z +<\# # I\$ % G9@\" \# " G9@\" \$ # G9@\# \ $ $ Find the credibility premium for period 3 in terms of \" and \# using Buhlmann's credibility ap
21.
You are given: (i) The prior distribution of the parameter has probability density function:
=
(ii)
bg
1 , 1< < 2
Given = , claim sizes follow a Pareto distribution with parameters = 2 and .
A claim of 3 is observed. Calculate the posterior probabili
10-11.
Use the following information for questions 10 and 11. (i) The claim count and claim size distributions for risks of type A are: Number of Claims 0 1 2 (ii) Probabilities 4/9 4/9 1/9 Claim Size 500 1235 Probabilities 1/3 2/3
The claim count and cla
CREDIBILITY - PROBLEM SET 3
Bayesian Credibility - Discrete Prior
1. Two bowls each contain 10 similarly shaped balls. Bowl 1 contains 5 red and 5 white balls (equally likely to be chosen). Bowl 2 contains 2 red and 8 white balls (equally likely to be cho
RISK MEASURES PROBLEM SET
1. Find U* and GX I* for each of the following loss distributions. (a) Exponential distribution with a mean of 10,000 (b) Pareto distribution with mean 10,000 and standard deviation of 12,247. (c) Lognormal with mean 10,000 and s
32.
You are given the following experience for two insured groups:
Group 1 2 Total Number of members Average loss per member Number of members Average loss per member Number of members Average loss per member
Year 1 8 96 25 113 2 12 91 30 111 3 5 113 20 1
CREDIBILITY - PROBLEM SET 1
Limited Fluctuation Credibility
1. The criterion for the number of exposures needed for full credibility is changed from requiring
\ to be within !&I\ with probability *, to requiring \ to be within 5I\ with probability
*& . Fi
CREDIBILITY - PROBLEM SET 5
Bayesian Credibility - Exam C Table Distributions
1. An individual insured has a frequency distribution per year that follows a Poisson distribution
with mean -. The prior distribution for - is exponential with a mean of 2. An
CREDIBILITY - PROBLEM SET 4
Bayesian Credibility - Continuous Prior
Problems 1 and 2 are based on the following situation. Assume that the number of claims each
year for an individual insured has a Poisson distribution. Assume also that the expected annua
CREDIBILITY - PROBLEM SET 1
Limited Fluctuation Credibility
1. The criterion for the number of exposures needed for full credibility is changed from requiring
\ to be within !&I\ with probability *, to requiring \ to be within 5I\ with probability
*& . Fi
CREDIBILITY - PROBLEM SET 6
Buhlmann Credibility
1. Assume that the number of claims each year for an individual insured has a Poisson
distribution. Assume also that the expected annual claim frequencies (the Poisson parameters A)
of the members of the po
CREDIBILITY - PROBLEM SET 3
Bayesian Credibility - Discrete Prior
1. Two bowls each contain 10 similarly shaped balls. Bowl 1 contains 5 red and 5 white balls
(equally likely to be chosen). Bowl 2 contains 2 red and 8 white balls (equally likely to be
cho
CREDIBILITY - PROBLEM SET 7
Empirical Bayes Credibility Methods
Problems 1 to 3 refer to the following situation. An insurance company has two group policies.
The aggregate claim amounts (in millions of dollars) for the first three policy years are
summar
ACT466H1S - TEST 1 - FEBRUARY 7, 2008
Write name and student number on each page. Write your solution for each
question in the space provided.
6
1. Suppose that events G and H are conditionally independent of events I and I w (complement
of I ). Derive th
ACT466H1S - TEST 2 - MARCH 22, 2007
Write name and student number on each page. Write your solution for each
question in the space provided.
1. The parameter : has a prior distribution with pdf 1: #: for ! : " .
The model distribution \ is a discrete non-
ACT466H1S - TEST 1 - FEBRUARY 8, 2007
Write name and student number on each page. Write your solution for each
question in the space provided.
1. A compound distribution W has a Poisson frequency distribution R with mean -.
For parts (a) and (b), assume t
10-11.
(Repeated for convenience) Use the following information for questions 10 and 11. (i) The claim count and claim size distributions for risks of type A are: Number of Claims 0 1 2 (ii) Probabilities 4/9 4/9 1/9 Claim Size 500 1235 Probabilities 1/3
29.
In order to simplify an actuarial analysis Actuary A uses an aggregate distribution S = X1 +.+ X N , where N has a Poisson distribution with mean 10 and X i = 15 for all i. . Actuary As work is criticized because the actual severity distribution is gi
ACT466H1S - TEST 1 - FEBRUARY 8, 2007 Write name and student number on each page. Write your solution for each question in the space provided.
1. A compound distribution W has a Poisson frequency distribution R with mean -. For parts (a) and (b), assume t
28.
The random variable X has the exponential distribution with mean Calculate the mean-squared error of X 2 as an estimator of 20 4 21 4 22 4 23 4 24 4
.
2
.
(A) (B) (C) (D) (E)
Exam C: Fall 2005
-28-
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