Practice_Exam_3-Questions

Practice_Exam_3-Questions - PRACTICE EXAMINATION NUMBER 3 1...

Info icon This preview shows pages 1–7. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: PRACTICE EXAMINATION NUMBER 3 1. An insurance company issues insurance contracts to two classes of independent lives, as shown below. Class Probability of Death Benefit Amount Number in Class A 0.01 200 500 B 0.05 100 300 The company wants to collect an amount, in total, equal to the 95-th percentile of the distribution of total claims. The company will collect an amount from each life insured that is proportional to that life’s expected claim. That is, the amount for life j with expected claim E(Xj) would be kE(XJ. ). Calculate k. A. 1.30 B. 1.32 C. 1.34 D. 1.36 E. 1.38 2. An actuary is reviewing a study she performed on the size of claims made ten years ago under homeowners insurance policies. In her study, she concluded that the size of claims followed an exponential distribution and that the probability that a claim would be less than $1000 was 0.25. The actuary feels that the conclusions she reached in her study are still valid today with one exception: every claim made today would be twice the size of a similar claim made ten years ago as a result of inflation. Calculate the probability that the size of a claim made today is less than 1000. A. 0.063 'B. 0.125 C. 0.134 D. 0.163 E. 0.250 3. A dental insurance policy covers three procedures: orthodontics, fillings and extractions. During the life of the policy, the probability that the policyholder needs: 0 Orthodontic work is 1/2. 0 Orthodontic work or a filling is 2/3. 0 Orthodontic work or an extraction is 3/4. 0 A filling and an extraction is 1/8. The need for orthodontic work is independent of the need for a filling and is independent of the need for an extraction. Calculate the probability that the policyholder will need a filling or an extraction during the life of the policy. A. 7/24 B. 3/8 C. 2/3 D. 17/24 E. 5/6 4. The value, v, of an appliance is based on the number of years since purchase, t, as follows: v(t) = e042” . If the appliance fails within seven years of purchase, a warranty pays the owner the value of the appliance. After seven years, the warranty pays nothing. The time until failure ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 172 - PRACTICE EXAMINATION NO. 3 of the appliance failing 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 B. 352.16 5. An automobile insurance company divides its policyholders into two groups: good drivers and bad drivers. For the good drivers, the amount of an average claim is 1400, with a variance of 40,000. For the bad drivers, the amount of an average claim is 2000, with a variance of 250,000. Sixty percent of the policyholders are classified as good drivers. Calculate the variance of the amount of a claim for a policyholder. A. 124,000 B. 145,000 C. 166,000 D. 210,400 B. 235,000 2+e' 9 6. Let X be a random variable with moment generating function M (t) =[ J . Calculate the variance of X. A.2 B.3 C8 D9 5.11 7. An insurance company designates 10% of its customers as high risk and 90% as low risk. The number of claims made by a customer in a calendar year is Poisson distributed with mean 0 and is independent of the number of claims made by a customer in the previous calendar year. For high risk customers 0 = 0.6, while low risk customers 9 = 0.1 . Calculate the expected number of claims made in calendar year 1998 by a customer who made one claim in calendar year 1997. A. 0.15 B. 0.18 C. 0.24 D. 0.30E. 0.40 8. Let X and Y be random losses with joint density function f” (x, y) = 2.x for 0 < x <1 and 0 < y < 1. An insurance policy is written to cover the loss X + Y. The policy has a deductible of 1. Calculate the expected payment under the policy. A. 1/4 B. 1/3 C. 1/2 D. 7/12 E. 5/6 9. Under a group insurance policy, an insurer agrees to pay 100% of the medical bills incurred during the year by employees of a small company, up to a maximum total of one million dollars. The total amount of bills incurred, X, has probability density function ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 173 - PRACTICE EXAMINATION N0. 3 x(4 - x) fx (x) = 9 ’ 0, otherwise. where x is measured in millions. Calculate the total amount, in millions of dollars, the insurer would expect to pay under this policy. for0<x<3, A. 0.120 B. 0.301 C. 0.935 D. 2.338 E. 3.495 10. Suppose the remaining lifetimes of a husband and a wife are independent and uniformly distributed on the interval (0, 40). An insurance company offers two products to married couples: 0 One which pays when the husband dies; and 0 One which pays when both the husband and wife have died. Calculate the covariance of the two payment times. A. 0.0 B. 44.4 C. 66.7 D. 200.0 B. 466.7 11. An insurance contract reimburses a family’s automobile accident losses up to a maximum of two accidents per year. The joint probability distribution for the number of accidents of a three person family (X, Y, 2) is p(x,y,z)= k(x+2y+z), where x= 0,1,y=0,1,2,z=0,l,2 , and x, y and z are the number of accidents incurred by X, Y and Z, respectively. Detemiine the expected number of unreimbursed accident losses given that X is not involved in any accidents. A. 5/21 B. 1/3 C. 5/9 D. 46/63 E. 7/9 12. The loss amount, X, for a medical insurance policy has cumulative distribution function 0, x<0, F(x)- 1 sz-x—3 0<x<3 X 9 3 ’ ' ‘ 1, x>3. Calculate the mode of the distribution. AZ B.1 Cg D.2 E3 3 2 13. A small commuter plane has 30 seats. The probability that any particular passenger will not show up for a flight is 0.10, independent of other passengers. The airline sells 32 tickets for the ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 174 - PRACTICE EXAMINATION N0. 3 flight. Calculate the probability that more passengers show up for the flight than there are seats available. A. 0.0042 B. 0.0343 C. 0.0382 D. 0.1221 E. 0.1564 14. LetX and Yhave a bivariate normal distribution with means p, = [1y = 0, variances 0': =1, 1 . . . . 0'; = 2 and correlation pxy = 2' What is the condrtronal variance of Y, given X = x? A B.3 Q] 13.2 E.2 4 2 l ' 2 15. Let X be a continuous random variable with density function lx3, for0<x<2, fx(x)= 4 0, otherwise. What is the density function of Y = 8 - X3 for 0 < y < 8? 1 l 1 ' 1 -2 _ 1 1 _ 1 .— — B — .— - D.—8— E.—8— A 3(8 y)3 4(8 y)3 C 6(8 y)3 12( y)3 4( y) 16. In a dice game, the player independently rolls a fair red die and a fair green die. The player wins if and only if the red die shows a l, 2, or 3, or if the total on the 2 dice is 11. What is the probability the player will win? 29 7 4 19 E_ _ 36 A. — B. — C. — D. 2 36 9 36 9 17. In a large population of people, 50% are married, 20% are divorced, and 30% are single (never married). In a random sample of 4 people, what is the probability that exactly 3 are married? [4] 3 E _ 4 3 4 4 3 4 A. 0.5 .o.2.o.3 B. 0.5 c. 0.5 D. 0.5 . 3 3 4! ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 175 - PRACTICE EXAMINATION N O. 3 18. Let X and Y be random variables whose joint distribution is uniform over the half-disc: {(Jc,y)lx2 + y2 S 1 and x 2 0}. What is the marginal density function of X for 0 S x S 1? A. $(1—x2fi B. 2(l—x2)% c. i-(l—xzfi D. g E. 1 19. An urn contains 10 balls: 5 are white, 3 are red, and 2 are black. Three balls are drawn at random, with replacement, from the urn. What is the probability that all 3 balls are different colors? A. 0.03 B. 0.09 C. 0.18 D. 0.40 E. 0.84 20. Three fair dice are tossed independently. Let E, denote the event that the i-th die results in a 6. What is Pr(E1U E, u E3)? E 125 1 .5. C21. ,_ 216 A. —- B. . 216 12 216 Nit—- 21. HE and F are events for which Pr(Eu F) = 1, then Pr(EC UFC)must equal A.0 B. Pr(EC)+Pr(FC)-Pr(EC)Pr(FC) C. Pr(EC)+Pr(FC) D. Pr(Ec)+Pr(Fc)—l E.1 22. What is the probability that a hand of 5 cards chosen randomly and without replacement from a standard deck of 52 cards contains the king of spades, exactly 1 other king, and exactly 2 queens? [414V] 000“] 0F”) fill“) [31“] 2 2 1 1 2 1 3 1 1 2 1 l 2 AT 3-5—2 QT ”—33— ET 5 ’5 5 5 5 23. Let X and Y be continuous random variables with joint density function ASM Study Manual for Course Pl] Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 1‘76 - PRACTICE EXAMINATION N0. 3 6xy+3x2, forO <x<y<1, 0, otherwise. xx.,(x.y)={ What is 15(le = y)? 2 2 3 A. 3y ,3_ zys C. 3&3: D6213: E, m 4y 4y 16 24. Let X be a discrete random variable with probability function Pr(X = x) = $- for x = l,2,3,.... What is the probability that X is even? A. l B. 3 c. l 13.3 E. 3 4 7 3 3 4 25. Let X1 ,X2, and X3 be independent continuous random variables each with density function JE-x, for0<x<\/§, f(x)= . 0, otherwrse. What is the probability that exactly 2 of the 3 random variables exceed 1? A. g-Ji B. 3—2J5 c. 3(~/§-1)(2—~/5)2 D-(E-fillf-él E-3li-J5Tlfi-él 26. Let the random variable X have moment generating function M(t) = e3”'1. What is E(X2)? A.l B.2 C.3 D.9 El] 27. Defective items on an assembly line occur independently with probability 0.05. A random sample of 100 items is taken. What is the probability that the first item sampled is not defective, given that at least 99 of the sampled items are not defective? A. ——5 B. E C. E D. 2 E. —5‘90 5.90 100 99 100 5.95 ASM Study Manual for Course Pl] Actuarial Examination. (9 Copyright 2004-2008 by Knysztof Ostaszewski - 177 - PRACTICE EXAMINATION NO. 3 , p 28. Let X and Ybe independent continuous random variables with common density function J l for0<x<l A (x) = {0, What is Pr(X2 2 Y3) ? otherwise. 29. Let X have a binomial distribution with parameters n and p, and let the conditional distribution of Y given X = x be Poisson with mean x. What is the variance of Y? A. x B. np C. np(1—p) D. up2 E. np(2 —p) 30. A sample of size 3 is drawn at random and without replacement from population {1, 2, 3, 4, 5}. What is the probability that the range of the sample is equal to 3'? A.—1— Bi oi v.3 5.31 15 125 10 10 5 ASM Study Manual for Course P/l Actuarial Examination. (0 Copyright 2004-2008 by Krzysztof Ostaszewski - 178 - .. J ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern