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Practice_Exam_7-Questions

Practice_Exam_7-Questions - PRACTICE EXAMINATION NUMBER 7 1...

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Unformatted text preview: PRACTICE EXAMINATION NUMBER 7 1. A population consists of 20% children, coded as 0 in a population database, 40% adult males, coded as l in the same population database, and 40% adult females, coded as 2 in the same database. A sample of two codes X1, X2, is chosen randomly and with replacement from this population’s database. Find Pr(|Xl - X2| = 1). A. 0.48 B. 0.32 C. 0.24 D. 0.16 E. 0.08 2. You are given that if a car gets into an accident, the dollar amount of damage has the gamma distribution with parameters a = 2 and fl = 0.001. The parameters, as used in this problem, have the following meaning: if a gamma distribution is obtained as a sum of random sample from an exponential distribition, then the parameter a is the one that counts how many elements the sample contains, and [3 is the hazard rate of the exponential distribution considered. What is the coefficient of variation of the loss modeled by this gamma distribution? A.— B Ci D] 13.2 1 10 ‘2 .5 3. There are two urns. The first one has 15 red balls and 5 black balls. The second one contains 14 red balls and 6 black balls. We choose randomly one of the urns, each of them equally likely to be chosen, and then we pick three balls from that urn (without replacement). It turns out that we have 3 red balls. Given that, what is the probability that we picked them from the second urn? [”1 R; C. i D E E 28 4o ' 29 ° 29 3 4. You are the actuary in charge of purchasing a reinsurance contract for your insurance company. You have determined that the losses that you want reinsured follow a uniform distribution on the interval [1000, 2000]. You have a choice of two reinsurance contracts for these losses. The first contract will pay 90% of the loss, while the second contract will pay up to a maximum limit, where the limit is set so that the expected payment for both contracts are the same. Find the ratio of the variance of the reinsurance payment under the second policy to the variance of the reinsurance payment under the first policy. A. 1.5 B. 1.2 C.0.9 D.0.6 E.0.3 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski - 289 - PRACTICE EXAMINATION NO. 7 5. An insurance company pays claims at a Poisson rate of 2,000 per year. Claims are divided into three categories: “minor,” “major,” and “severe,” with payment amounts of $1,000, $5,000, and $10,000, respectively. The proportion of “minor" claims is 50%. The total expected claim payments per year are $7,000,000. What is the proportion of claims that are “severe”? A. Less than 11% B. At least 11%, but less than 12% C. At least 12%, but less than 13% D. At least 13%, but less than 14% E. 14% or more 6. Let X be a random variable representing the number of times you need to roll (including the last roll) a fair six-sided dice until you get 4 consecutive 6's. Find E(X). A. 125 B. 1024 C. 1554 C. 2048‘ D. 15447 7. Let X be an exponential random variable with mean 8 and let Z be a Bernoulli Trial with probability of success p = 0.55. Calculate Var(ZX). A. 35.2 B. 51.04 C. 52.36 D. 64.00 B. 128.00 8. A delay in departure of a single plane from an aiport has the probability of %, with each individual departure treated as a Bernoulli Trial. Find the probability that at least 40 out of 180 planes will be delayed, using the normal approximation with continuity correction. A. 0.0345 B. 0.0288 C. 0.0197 D. 0.0110 E. 0.0096 9. A random variable X has a distribution with mean 500 and standard deviation 150, and its distribution is symmetric about its mean. Find the best approximation for Pr(X > 800) given by the Chebyshev’s Inequality. A. 0.0675 B. 0.1025 C. 0.1250 D. 0.2500 E. 0.3725 10. Let X and Y have a bivariate normal distribution with E (X) = px = 2, E (Y) = p, = —3, Var(X) = a; = 4, Var(Y) = of. = 25, and the correlation coefficient Px.r = —0.3. What is ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski - 290 - PRACTICE EXAMINATION NO. 7 qusflY=07 A. 0.54 B. 0.59 C. 0.64 D. 0.71 E. 0.78 11. A card is drawn from a deck, not replaced, and a second card is drawn. What is the probability that the second card is a heart? A. —3- B. E C 11 D ll 13 . — .— E 51 52 51 l ’4 12. A device contains two circuits. The second circuit is a backup for the first, i.e., the second circuit is used only when the first one has failed. The device fails when, and only when, the second circuit fails. Let X and Ybe the times at which the first and second circuits fail, respectively, counting from the moment when a circuit is activated. This means, effectively, that X and Y are the lifespans of the first, and the second circuit, respectively. X and Y are independent and identically uniformly distributed on the unit interval from 0 to 1 hour each. What is the probability that the device fails within 30 minutes? A.— B.— C.— D E l l 64 32 16 ‘8 ' 4 13. Assume that the number of train delays during a year follows a Poisson distribution with the mean of 10. Find the probability that the time between the next two delays is at least 3 months. A. 0.082] B. 0.1112 C. 0.2500 D. 0.3031 E. 0.3679 14. Three cards are drawn from a standard deck. What is the probability that all three are hearts, given that at least two of them are hearts? A. 0.0859 B. 0.0781 C. 0.07134 D. 0.0625 E. 0.0576 15. An American male 100 meters runner is preparing for the 2012 Olympic Games. After repeated runs, he determines that his 100 meters sprint run time is normally distributed with mean 9.88 seconds and standard deviation 0'. He has been also observing his Russian competitor and determined that the Russian’s sprint time is normally distributed with mean 9.99 seconds, and the same standard deviation 0'. Given that the probability that the American beats the Russian is 0.9015, and that their sprint times are independent, determine 0'. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Osmszewski - 291 - PRACTICE EXAMINATION N0. 7 A. 0.2512 B. 0.1025 C. 0.0875 D. 0.0603 E. 0.0499 16. You are a new pricing actuary for a Midwest Corn Fields Property and Casualty Insurance Company (MCFP&CCo). Existing homeowners‘ insuarance has been priced assuming that there is 0.90 probability of no loss, and 0.10 probability of a loss, with the actual loss amount, once a loss occurs, uniformly distributed between $0 and $500,000. The policy has a deductible of $5,000. Upon your arrival you determine that the model for the loss, once it occurs, is incorrect, and that it should have been assumed that such loss is uniformly distributed between $0 and $600,000. Find the amount by which the expected payment under your corrected pricing exceeds the expected payment under the old pricing scheme. A. 1,000 B. 2,000 C. 3,000 D. 4,000 B. 5,000 17. Given that A and B are events such that Pr(A) = Pr(B) =§ and Pr(AIB) = «é, find Pr(AuB). 11 5 2 A. — B. - C. — 6 3 18 9.3 El 9 2 18. The number of warranty claims in a month follows a Poisson distribution with mean 4. Each warranty claims requires a payment of $1200 by the manufacturer to a repair facility. Find the probability that the total payment by the manufacturer in a month is less than one standard deviation away from the expected value of that payment. A. 0.1225 B. 0.3907 C. 05470 D. 0.6936 E. 0.7977 19. Mr. Warrick Beige is playing blackjack in Las Vegas. His probability of winning at each game is 0.45, probability of losing is 0.50, and the probability of breaking even is 0.05. At every game, he either wins $10, loses $10, or breaks even, and he plays 100 times. Using normal approximation, estimate the probabillity that he actually comes out ahead in those 100 games. A. 0.10 B. 0.20 C. 0.30 D. 0.40 E.O.50 20. Find the value of the probability generating function P” (t) of the Poisson random variable N withmean2at t=%. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski - 292 - PRACTICE EXAMINATION NO. 7 A. 27.0434 B. 7.3891 C. 2.7183 D. 1.6487 E. 0.3679 21. A survey of customers of a property-casualty insurance company indicates that 20% of them have no children, 20% have one child, and 60% have two or more children. The same survey also shows that 30% of customers with no children own their home, 40% of customers with one child own their home, and 50% of the customers with two or more children own their home. Find the overall percentage of customers that own their home. A. 52% B. 48% C. 44% D. 40% E. 36% 22. An insurance company has determined that the annual claim amount distribution for its . . . . l . customers 18 exponential. The expected claim amount 18 a number C that E; has the uniform distribution on the interval [0 .001 , 0.002]. An insured is chosen randomly. Find the probability that this insured has annual claims in excess of 1000. A. 0.0012 B. 0.2325 C. 0.5011 D. 0.7670 E. 0.9991 23. The number of claims N per year in a certain reinsurance policy has been believed to follow a Poisson distribution with a mean of 1. Based on the reinsurer’s historical data, the 'reinsurer decides to use a new probability random variable M to describe the number of claims, with Pr(M = 0) = 0.5, and Pr(M = k) = c . Pr(N = k) for k =1, 2, with c being a certain constant. Find E(M). A. 0.79 B. 0.63 C. 0.50 D. 0.37 E. 0.21 24. At a certain insurance company the weights of male and female employees are normally distributed: N (200.2500) for males, and N (140,1600) for females (all weights are in pounds, and N ( {1,02) stands for a normal distribution with mean u and variance 0'2 ). A group of ten employees, five males and five females, is being flown from the company headquarters to an office in Iowa in a small company plane, with the plane’s maximum load for passengers being 2000 pounds. What is the probability that the total weight of the passengers exceeds the 2000 pounds limit, assuming that weights of individual employees are independent random variables? A. 0.0668 B. 0.0526 C. 0.0409 D. 0.0250 E. 0.0181 25. A certain family health insurance policy covers up to two claims per person during a year. In ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 2004-2008 by Krzysztof Ostaszewski - 293 - PRACTICE EXAMINATION N0. 7 a certain family, there are three family members, a male adult (father), a female adult (mother), and a child. The joint probability function for the number of claims in a year is 6—x-y—z fx.r.z (MW) = T where x, y, and 2 can assume values of 0, 1, 2, and X, Y, and Z are the numbers of claims for the male adult, female adult and the child members of the family. Find the probability that the total number of claims for the entire family in a year does not exceed 2, given that the male adult member of the family did not have any claims for that year. ’ A B.1 C.-5- D.1 E.— 3 6 9 l ' 2 26. Let (X ,Y) have the joint density function 6(1—x—y), forOSxSl,0SySlandOSx+yS1, fxm (x,y) = . 0, otherw15e. What is Pr[0 S X S i]? A.— 3.1 C.— D E]- 8 8 l 12 12 ° 4 27. A point (X ,Y) is chosen at random from the unit disk at2 + y2 51 according to a uniform distribution on the disk. What is the value of f, (0.25IY = 0.75)? 2 3.4. 7 J7 A. 0 B D. l 4 .— C.— 7: m/7 28. A family has five children. Assuming that the probability of a girl on each birth was 0.5 and that the five births were independent, what is the probability the family has at least one girl, given that they have at least one boy? 31 30 15 5 5 A. —— B. — C. — . . 32 31 16 31 32 29. Let X1, X2 , and X3 be a random sample from a distribution with density function ASM Study Manual for Course PI] Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 294 - v ‘ PRACTICE EXAMINATION NO. 7 W 2):, forO<x<l, fx(x)= 0, otherwise. Let Y0) S Y(2) S Y0) be the order statistics for this random sample. What is E (Ym)? 1 J3 2 3 6 A. — B. —— c. - D. — E. — J3 2J5 3 4 7 30. A calculator has a random number generator button which, when pushed, displays a random digit (0, 1, 2, .. . , 9). The button is pushed four times. Assuming the numbers generated are independent, what is the probability of obtaining one “0”, one “5”, and two “9”’s in any order? 10! 1 ‘° 101(1)“ 41(1)‘ 91(1)‘ A.—- — B.—- - c.—- —- D.—- — 21 [1o] 2! 10 21 1o 4! 9 (a... ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20044008 by Knysztof Ostaszewski - 295 - ...
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