Practice_Exam_16-Questions

Practice_Exam_16-Questions - PRACTICE EXAMINATION NUMBER 16...

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Unformatted text preview: PRACTICE EXAMINATION NUMBER 16 1. X has a uniform distribution on the interval [0,10]. Find the hazard rate of X at 7.50. A. O B. 0.04 C. 0.25 D. 0.40 B. 0.50 2. Let X denote the number of independent rolls of a fair die required to obtain the first "3" . What is Pr(X .>. 6)? 3. Let X be a continuous random variable with density function axze'b‘ for x > 0, fx (x) = {0 elsewhere, where a > 0 and b > 0. What is the mode ofX? A.0 B.2 C. 3 D. 9- E. 00 b 2 4. Let X be a random variable with finite variance. If Y = 15 - X, then the correlation coefficient ofX and (X+Y)X equals A. -l B. 0 C. i D. 1 E. Cannot be determined from the 15 information given 5. Let X and Y have a bivariate normal distribution with means px = 5 and p, = 6, standard deviations ox = 3 and a, = 2, and covariance 0x, = 2. Let (1) denote the cumulative distribution function of a normal random variable with mean 0 and variance 1. What is Pr(ZX-Y $5) in terms of (I)? ”(J—z.) we) c- it?) we) we.) ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 2004-2008 by Knysztof Ostaszewski - 551 - SECTION 20 6. A fair coin is tossed. If a head occurs, 1 die is rolled; if a tail occurs, 2 dice are rolled. Let Y be VJ] the total on the die or dice. What is E (Y)? 7 21 21 A. — B. 5 C. —- D.7 E. — 2 4 2 7. An um contains 2 white and 8 red marbles. A marble is drawn from the urn 100 times in succession with replacement. Which of the following is closest to the probability of drawing more than 75 red marbles? A. 0.11 B. 0.62 C. 0.75 D. 0.87 E. 0.95 8. A system has 2 components placed in a series so that the system fails if either of the 2 components fails. The second component is twice as likely to fail as the first. If the 2 components operate independently, and if the probability that the entire system fails is 0.28, then what is the probability that the first component fails? A. 23—8 B.0.10 C. % D.0.20 E. ~10.14 9. Let Z‘ , Z2, Z3 be independent normal random variables each with mean 0 and variance 1. Which of the following has a chi-square distribution with 1 degree of freedom? 2 2 AL}: B. (z,+z,)’—z,2 c.zf+z§—z32 2 D. M E, (21 +z2—23Y 10. A pair of dice is tossed 10 times in succession. What is the probability of observing no 7’s and no 11’s in any of the 10 tosses? Ag)” Ber-e)” atlas-1') ”14%)”. .0 E. [14%) W l ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Osmszewski - 552 - PRACTICE EXAM 16 11. The joint probability density for X and Y is 260””), forx > 0,y > o, x, = fx'Y( y) {0, otherwise. Calculate the variance of 1’ given that and X > 3 and Y> 3. A. 0.25 B. 0.50 C. 1.00 D. 3.25 E. 3.50 12. Three boxes are numbered 1, 2, and 3. For k = 1, 2, 3, box k contains k blue marbles and 5 — kred marbles. In a two-step experiment, a box is selected and 2 marbles are drawn from it without replacement. If the probability of selecting box kis proportional to k, what is the probability that the 2 marbles drawn have different colors? 17 34 l 8 17 A. — B. — C. D. — E. — 60 75 2 15 30 13. The probability that a property will not be damaged in the next period is 0.80. Moreover, if the property is damaged, given that the damage occurs, the probability density function (PDF) of the amount of loss is given by fx (x) = 0.01e’°”" for x > 0. Calculate the standard deviation of the loss X, where X = 0 if no damage occurs, or X is the actual amount of damage, if the property under consideration is damaged. A. 20 B. 40 C. 60 D. 80 E. 100 14. Mr. Warrick Beige plays a game at the famous You Was Robbed casino. In the game, Mr. Beige must pay $200 to enter the game, and then a coin is tossed and Mr. Beige is paid $250 -(1- 2'x ), where X is the number of the first toss that results in a head. The coin used by the casino is assumed by the players to be fair, but it is not. Its probability of tails is 0.60, and for heads the probability is 0.40. Find the expected value of the difference between the amount paid by Mr. Beige and the payout he receives. A. —$50 B. $0 C. $10 D. $20 E. $50 15. You are running a small business, which faces a risk of damage to its equipment, with the probability distribution of the loss amount X (in thousands) having the density fx (x) = 0.1e'°"" for x > 0, and 0 otherwise. You are considering a purchase of an insurance policy to cover that loss. You can buy a policy covering the whole loss for the premium equal to E (X) , but you realize that you cannot afford it and instead purchase a policy that will pay nothing if the loss is under 101n2 (in thousands), and X — 101n2 (in thousands) if the loss is above 101n2 (in thousands). Calculate the savings in premium (in thousands) versus the purchase of full ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 2004-2008 by Knysztof Ostaszewski - 553 - SECTION 20 coverage, if the second policy can also be obtained for the premium equal to the expected value of the amount of loss paid by the insurance firm. A.l B.ln2 C. e D.4 ES 16. Let A, B, and C be three events such that Pr(A|c) = 0.05, Pr(B| c) = 0.05, and A and B are mutually exclusive. If Pr(A u B) = Me) = 0.80, what is Pr(C| A u B)? A.0.05 B.0.10 C.0.15 D.0.20 E.0.25 17 . A student in a probability class sends an e-mail to her professor teaching the class. One out of every thousand e-mails is destroyed by a computer virus planted in the computer system by a hacker. Assuming the professor is Polish, and thus required by the customs of Polish culture to answer every e—mail received, what is the probability that the student’s e-mail did not reach the professor, given that the student does not receive a response? Assume that disappearances of messages are independent of each other. A. 0.4900 B. 0.4975 C. 0.5000 D. 0.5003 E. 0.6025 18. Let X be the number of heads observed in four tosses of a fair coin. Given the value of X, exactly X fair six-faced dice, independent of each other, are thrown. Let Ybe the sum of the numbers showing on the dice. Find the coefficient of variation of Y. A. 0.3333 B. 0.4875 B. 0.6075 C. 1.3333 E. 2.1251 19. Mr. Soichiro Gondo has started a computer game company, and has purchased an office building to house the headquarters of his firm. The probability that a fire occurs at his office building during the upcoming year is 0.01 . Given a fire, the damage is uniformly distributed over the interval between zero yen and one million yen. Calculate the (unconditional) standard deviation of the fire damage to Mr. Gondo’s office building during the upcoming year. A. 9925 B. 10000 C. 29533 D. 57518 E. 102888 1 50 20. You are given the moment generation function of a random variable Y, M), (t) = [l—t] , for t < 1. Use the normal approximation, with continuity correction, if needed, to estimate the 90- th percentile of Y. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Krzysztof Ostaszewski - 554 - PRACTICE EXAM 16 A. 51 B. 53 C. 55 D. 57 E. 59 21. 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 information is known aboutX 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 The the total number of hours that these one hundred people watch movies or sporting events during this three-month period. Approximate the value of Pr(T < 7100). A. 0.62 B. 0.84 C. 0.87 D. 0.92 E. 0.97 22. Let X have a uniform distribution on the interval (1, 3). What is the probability that the sum of 2 independent observations of X is greater than 5? A.— B.1 C D E.—5- 8 8 l l .4 .2 23. You are given the joint distribution of X and Y as defined by the probability density function _ 1, —l<x<l,|x|<y<1, fx‘y(1.y) - {0. otherwise. Find the coefficient of variation of X + Y. A.—l— B #3 .1 B.l DE E.— 3 2 3 24. Let X and Y be independently distributed normal random variables with means #x = 3 and p, = 5 and variances a; = 9 and 0'}. =16. Which of the following is closest to the probability that Y — X is greater than 7? A. 0.03 B. 0.16 C. 0.42 D. 0.84 E. 0.97 25. You are given that X and Y have the uniform distribution over the region R in the xy-plane JX+J§7 defined by the conditions 0 < x < l and 0 < 2y < x. Calculate the expected value of e . J X Y A. 3.70 B. 7.14 C. 14.81 D. 29.62 E. 59.24 ASM Study Manual for Course P/l Actuarial Examination. (0 Copyright 2004-2008 by Knyszlof Osmszewski - 555 - SECTION 20 26. You are given that a random variable N is discrete and assumes only positive integer values W») with Pr(N = n) proportional to 2‘" for n = 1,2,3,. . .. Find the expected value of N. A B. l C. l:- D. 2 E. Does not exist 1 ' 2 27. X is an exponential random variable with hazard rate 1. Find E ((X - 1)3| X 2 E (X )) A. 1.00 B. 2.21 C. 2.71 D. 3.00 E. 6.00 28. A ball is drawn at random from a box containing 10 balls numbered sequentially from 1 to 10. Let X be the number on the ball selected, let R be the event that X is an even number, let S be the event that X 2 6, and let The the event that X S 4. Which of the pairs (R,S), (R,T), and (S,T) are independent? A. (R,S) only B. (R,T) only c. (S,T) only D. (R,S) and (R,T) only E. (R,S), (R,T), and (S,T) 29. T1 and T2 are two independent, identically distributed, exponential random variables with mean 1. Find the joint density function of X = T, + 2T2 and Y = T1. 1 -1(x+y) 1 -l(x+y) 1 -l(x+y) A. Eel forx>0andy>0 B. 5e? forx>y>0 C. Eel fory>x>0 l D. e 20‘ Y) forx>0andy>0 E. e‘w’) forx>0andy>0 30. X and Y are two random variables whose moment generating functions are related by the following formula: anMy (t) = Mx(t)- 1. You are given that E(X) = 0 and Var(X) = 4. Find Var(Y). A. 1 11.5 C4 D.8 E. 11164 ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20044008 by Krzysztof Ostaszewski - 556 - ...
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