stat 477 name: winter 2016 practice exam 1

Stat 477 Name:Winter 2016Practice Exam 102 February 2016

This exam contains 13 pages (including this cover page)and 6 problems. Check to see if any pages are missing.

You may only use an SOA-approved calculator and apencil or pen on this exam.

You are required to show your work on each problem onthis exam.

Problem Points Score

1 10

2 13

3 5

4 15

5 4

6 6

Total: 53

Stat 477 Practice Exam 1 - of 13 02 February 2016

1. Suppose you observe 6 claims of size 10, 15, 15, 15, 20, and 40. Using the empirical distribution(assuming that each claim is equally likely), calculate

(a) (2 points) E(X)

(b) (2 points) E(X2)

(c) (3 points) E((X − 17)+)

(d) (3 points) E((X ∧ 17))


Mostly blank page to be used as extra space if needed.


2. The loss random variable X has pdf

f(x) =1

3(1 + 2x)e−x, for x > 0

Hint:∫ze−zdz = −(z + 1)e−z

(a) (5 points) Calculate E(X).

(b) (6 points) Calculate e(2).

(c) (2 points) Explain the difference between E(X) and e(2) as you would to senior manage-ment.


3. Let X be a random variable with the following loss distribution

k pk0 0.65100 0.30200 0.03500 0.011000 0.01

(a) (2 points) Calculate the 90% value-at-risk.

(b) (3 points) Calculate the 90% tail value-at-risk.


4. Assume that for a certain block of personal auto business, the number of claims in a weekfollow a Poisson distribution and average about 1 a day. Any given claim has a 0.10 probabilityof being fraudulent.

(a) (2 points) How many fraudulent claims are expected in one year (52 weeks)?

(b) (4 points) Prove the above expected value using nothing more than the density function

pk =e−λλk

k!.

(c) (2 points) What is the probability that there are no fraudulent claims in three consecutiveweeks?

(d) (3 points) Given there were 3 claims in one week, what is the probability that none ofthem are fraudulent?

(e) (4 points) Given there were r claims in one week, what is the distribution of the numberof fraudulent claims?


5. (4 points) Suppose that the number of claims follows a zero-modified Poisson(2) distribution.Under this distribution, the mean number of claims is 1.50. Calculate the probability that thenumber of claims will be greater than one.


6. (6 points) Assume that the number of claims follows a Bin(11, q) distribution. Assume furtherthat q follows a Beta(3, 5) distribution. The pdf of a Beta(a, b) distribution is

f(x) =Γ(a+ b)

Γ(a)Γ(b)xa−1(1− x)b−1

Find the probability that the number of claims is exactly 1.

stat 477 name: winter 2016 practice exam 1

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