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Math 370, Actuarial Problemsolving
Spring 2008
A.J. Hildebrand
Practice Test, 1/28/
About this test. This is a diagnostic test made up of a random collection of 20 problems from past Course 1/P actuarial exams. It is intended for students who have already taken of Math 408 or equivalent (e.g., Math 461), to give them an idea of where they stand. It also helps me identify areas that should be emphasized in this course. This test will not count, so if you do poorly, it won’t affect your grade. If you want, you can take the test anonymously—just leave out your name on the answer sheet. However, in order for the test to provide meaningful feedback, I’d like everyone who is taking the test to turn in an answer sheet.
Note for those currently enrolled in 408: If you have not already taken 408, it does not make much sense to take this test, since most of the problems are on material not yet covered in 408. Instead work the set-theory practice problems.
Rules. This test is intended to simulate the real thing as closely as possible, so you should abide by the same rules. In particular, no notes, books, etc., and use calculators only for basic arithmetic operations. In particular, do not use calculators to compute integrals, derivatives, or to plot graphs. In the actuarial exam you are limited to calculators without such functions. It goes without saying that you shouldn’t cheat. Don’t copy an answer from your neighbor; if you do so, you are only cheating yourself, and you are defeating the purpose of this course.
Time. You have 2:00 hours for this exam. This works out to 6 minutes per problem, which is the same as what you get in an actuarial exam. Use the time wisely, and don’t let yourself get bogged down in a lengthy calculation. If you don’t get a problem at first, move on to the next one.
Answers/solutions. I will post an answer key and partial solutions on the course webpage, www.math.uiuc.edu/∼hildebr/370.
Math 370 A.J. Hildebrand Spring 2008
- An actuary studying the insurance preferences of automobile owners makes the fol- lowing conclusions:
(i) An automobile owner is twice as likely to purchase collision coverage as disability coverage. (ii) The event that an automobile owner purchases collision coverage is independent of the event that he or she purchases disability coverage. (iii) The probability that an automobile owner purchases both collision and disability coverages is 0.15.
What is the probability that an automobile owner purchases neither collision nor disability coverage?
(A) 0. 18 (B) 0. 33 (C) 0. 48 (D) 0. 67 (E) 0. 82
Math 370 A.J. Hildebrand Spring 2008
- An insurance company pays hospital claims. The number of claims that include emergency room or operating room charges is 85% of the total number of claims. The number of claims that do not include emergency room charges is 25% of the total number of claims. The occurrence of emergency room charges is independent of the occurrence of operating room charges on hospital claims. Calculate the probability that a claim submitted to the insurance company includes operating room charges.
(A) 0. 10 (B) 0. 20 (C) 0. 25 (D) 0. 40 (E) 0. 80
Math 370 Practice Test, 1/28/2008 Spring 2008
- The value, ν, of an appliance is given by ν(t) = e^7 −^0.^2 t, where t denotes the number of years since purchase. If appliance fails within seven years of purchase, a warranty pays the owner the value of the appliance at the time of failure. After seven years, the warranty pays nothing. The time until failure of the appliance has an exponential distribution with mean 10. Calculate the expected payment from the warranty.
(A) 98. 70 (B) 109. 66 (C) 270. 43 (D) 320. 78 (E) 352. 16
Math 370 Practice Test, 1/28/2008 Spring 2008
- A company offers a basic life insurance policy to its employees, as well as a supplemen- tal life insurance policy. To purchase the supplemental policy, an employee must first purchase the basic policy. Let X denote the proportion of employees who purchase the basic policy, and Y the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f (x, y) = 2(x + y) on the region where the density is positive. Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the supplemental policy?
(A) 0. 010 (B) 0. 013 (C) 0. 108 (D) 0. 417 (E) 0. 500
Math 370 A.J. Hildebrand Spring 2008
- An auto insurance policy will pay for damage to both the policyholder’s car and the other driver’s car in the event that the policyholder is responsible for an accident. The size of the payment for damage to the policyholder’s car, X, has a marginal density function of 1 for 0 < x < 1. Given X = x, the size of the payment for damage to the other driver’s car, Y , has conditional density of 1 for x < y < x + 1. If the policyholder is responsible for an accident, what is the probability that the payment for damage to the other driver’s car will be greater than 0.500?
(A) 3/ 8 (B) 1/ 2 (C) 3/ 4 (D) 7/ 8 (E) 15/ 16
Math 370 A.J. Hildebrand Spring 2008
- The time, T , that a manufacturing system is out of operation has cumulative distri- bution function F (t) =
t
for t > 2, 0 otherwise. The resulting cost to the company is Y = T 2. Determine the density function of Y , for y > 4.
(A) 4y−^2 (B) 8y−^3 /^2 (C) 8y−^3 (D) 16y−^1 (E) 1024y−^5
Math 370 Practice Test, 1/28/2008 Spring 2008
- Let X and Y be the number of hours that a randomly selected person watches movies and sporting events, respectively, during a three-month period. The following infor- mation is known about X and Y :
E(X) = 50, E(Y ) = 20, Var(X) = 50, Var(Y ) = 30, Cov(X, Y ) = 10.
One hundred people are randomly selected and observed for these three months. Let T be the total number of hours that these one hundred people watch movies or sporting events during this three-month period. Approximate the value of P (T < 7100).
(A) 0. 62 (B) 0. 84 (C) 0. 87 (D) 0. 92 (E) 0. 97
Math 370 Practice Test, 1/28/2008 Spring 2008
- A company is reviewing tornado damage claims under a farm insurance policy. Let X be the portion of a claim representing damage to the house and let Y be the portion of the same claim representing damage to the rest of the property. The joint density function of X and Y is
f (x, y) =
6(1 − (x + y)) for x > 0 , y > 0 , x + y < 1, 0 otherwise.
Determine the probability that the portion of a claim representing damage to the house is less than 0.2.
(A) 0. 360 (B) 0. 480 (C) 0. 488 (D) 0. 512 (E) 0. 520
Math 370 A.J. Hildebrand Spring 2008
- The distribution of loss due to fire damage to a warehouse is:
Amount of loss
Probability
Given that a loss is greater than zero, calculate the expected amount of the loss.
(A) 290 (B) 322 (C) 1, 704 (D) 2, 900 (E) 32, 222
Math 370 A.J. Hildebrand Spring 2008
- Two insurers provide bids on an insurance policy to a large company. The bids must be between 2000 and 2200. The company decides to accept the lower bid if the two bids differ by 20 or more. Otherwise, the company will consider the two bids further. Assume that the two bids are independent and are both uniformly distributed on the interval from 2000 to 2200. Determine the probability that the company considers the two bids further.
(A) 0. 10 (B) 0. 19 (C) 0. 20 (D) 0. 41 (E) 0. 60
Math 370 Practice Test, 1/28/2008 Spring 2008
- The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The manufacturer agrees to pay a full refund to a buyer if the printer fails during the first year following its purchase, and a one-half refund if it fails during the second year. If the manufacturer sells 100 printers, how much should it expect to pay in refunds?
(A) 6, 321 (B) 7, 358 (C) 7, 869 (D) 10, 256 (E) 12, 642
Math 370 Practice Test, 1/28/2008 Spring 2008
- An insurance policy is written to cover a loss X where X has density function
f (x) =
3 8 x
(^2) for 0 ≤ x ≤ 2, 0 otherwise.
The time (in hours) to process a claim of size x, where 0 ≤ x ≤ 2, is uniformly distributed on the interval from x to 2x. Calculate the probability that a randomly chosen claim on this policy is processed in three hours or more.
(A) 0. 17 (B) 0. 25 (C) 0. 32 (D) 0. 58 (E) 0. 83
Math 370 A.J. Hildebrand Spring 2008
- In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years. The difference between the true age and the rounded age is assumed to be uniformly distributed on the interval from − 2 .5 years to 2.5 years. The healthcare data are based on a random sample of 48 people. What is the approximate probability that the mean of the rounded ages is within 0.25 years of the mean of the true ages?
(A) 0. 14 (B) 0. 38 (C) 0. 57 (D) 0. 77 (E) 0. 88
