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Practice_Exam_17-Questions

Practice_Exam_17-Questions - PRACTICE EXAMINATION NUMBER 17...

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Unformatted text preview: PRACTICE EXAMINATION NUMBER 17 1. A card hand selected from a standard deck consists of 2 kings, a queen, a jack, and a ten. Three additional cards are selected at random and without replacement from the remaining cards in the deck. What is the probability that the enlarged hand contains at least three kings? . 3 B. 132 C. 135 D. 264 E. 267 1081 1081 1081 1081 1081 2. Let Y have a uniform distribution on the interval (0,1), and let the conditional distribution of X given y be uniform on the interval (0,Jy). What is the marginal density function of X for 0 < x < l? l 1 1 A.21—x 3.2.x C.2 l-x‘ D.—--l E.— ( ) [ J J; 2J2 3. For each of its insured clients, losses insured by Courageous Property & Casualty Insurance in one year follow a Pareto distribution with 0 = 5000 and a = 1.2. - Xl ,X2,...,X5 represent losses from five independent policies. 0 Y, ,Y2,...,Ys are the order statistics associated with Xl ,X2,...,Xs. Calculate the probability that Y5 is greater than $25,000. A. Less than 20% B. At least 20%, but less than 30% C. At least 30%, but less than 40% D. At least 40%, but less than 50% E. At least 50% 4. You are given a random variable X with exponential distribution with mean 2. Let K = IIX]]. Find the coefficient of variation of K. A. 0.6065 B. 0.7788 C. 1.1331 D. 1.2840 E. 1.6487 5. Let Pr(AnB)=0.2, Pr(A) =O.6, and Pr(B) = 0.5. Then Pr(A° UBC)= A.0.1 B.0.3 C. 0.7 D.0.8 E.0.9 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Kraysztof Ostaszewski - 578 - PRACTICE EXAM 17 6. Let X be a Poisson random variable with mean A. If Pr(X =1|X S l) = 0.8, what is the value of 11? A.4 B. -1n.2 C.0.8 D.0.25 E. -1n .8 7. Mr. Wanick Beige is tossing a biased coin he obtained from the infamous You Was Robbed! casino. The probability of getting heads in each toss is 0.30. Let X be the number of heads in 400 independent tosses. Calculate the probability that X is bounded between 100 and 140 using the Chebyshev's Inequality. A. 0.21 B. 0.30 C. 0.50 D. 0.70 E. 0.79 :+-:’ 8. The moment generating function of a continuous random variable X is e 1 . Find Pr(X > 2). A. 0.8413 B. 0.5000 C. 0.1587 D. 0.0794 E. 0.0228 9. You are given the joint moment generating function of two random variables, X and Y: 1 M ,=— ”(so l-2s-3t+6st 1 1 for s<E and “3' Find Pr(min(X,Y)>0.95). A. 0.05 B. 0.10 C. 0.43 D. 0.45 E. 0.50 10. You are given that the joint density of random variables X and Y is 2 2 yx +xy—y , 0<y<x<l, fX.Y (LY) = ( ) . 0, otherw15e. Find the expected value of Y. A. 0.75 B. 0.50 C. 0.46 D. 0.33 E. 0.06 11. Claim sizes follow an exponential distribution with mean 200. A random sample of size 4 is drawn. Calculate the probability that the smallest claim will be larger than 100. A. 0.1353 B. 0.3679 C. 0.6667 D. 0.8825 E. 0.9500 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski - 579 - SECTION 21 12. Mr. Wat-rick Beige is examining a biased coin he obtained from the infamous You Was Robbed casino. The coin turns out to be biased with the probability of getting tails in each toss equal to 0.375. Find the mode of the distribution which counts the number of times you have toss this coin to get the third tails. A. 4 B. 5 C. 6 D. 10 E. Does not exist 13. The moment generating function of X and Y is M x J (3,!) = 0.2e’ (1 + 3e' + e’”'). Find the correlation coefficient of X and Y. A. 0.79 B. 0.82 C. 0.85 D. 0.89 E. 0.90 14. County Plantation Insurance Company offers snow insurance, which pays nothing if the daily snow fall is below 2 inches, and for higher level of snow fall it pays an increasing amount, changing in a linear fashion from $0 for 2 inches of snow to a maximum of $2000, with $500 paid for every additional 2 inches of snow above 2 inches. You are given that the amount of snow falling in a given day during the policy term follows an exponential distribution with hazard rate 2. Find the expected value of the amount of the claim under this snow insurance policy. A. 2.29 B. 3.00 C. 3.25 D. 6.66 E. 4.01 15. You are given that the joint distribution of X and Y is described by the probability function C(S — y) fory = 0.1.2.3,4,5 andx = 0,1,...,y, (w) ={ 0 otherwise, where C is a constant. Calculate variance of X. A.1 13.2 cfl 13.2 3.2 35 35 35 77 16. Santa Claus carries toys around the world in bags containing 10 toys each. Santa randomly examines 3 toys from a bag and approves the bag for delivery to children around the world if at least two of the toys have no defects. What is the probability that Santa will approve a bag if that particular bag contains exactly 2 defective toys? A; B.i cl 9.5 E.l—19- 120 15 15 15 120 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski - 580 . ’ PRACTICE EXAM 17 (w, 17. You are given a discrete random variable N such that its only possible values are 0, l, 2, 3, 4, and 5, and f” (n +1) > f” (n) for n = 0,1, 2, 3, 4. Additionally, you are given that fN("+l)‘fN(")=f/v(n+2)"f~("+1) for n = 0, 1, 2, 3. Finally, you also know that fN (0)+ fN (1) = 0.4. Find f" (4)+f~ (5). A.— B .1 13.1 4 15 .1 c.1 D 15 8 5 18. You are given that a random variable X is the sum of a random sample of size 3 from an exponential distribution with hazard rate 3. Find the moment generating function of the random variable %X. _ resents the relative fre- uenc of accidents -er da in a city. ————“- Relative 0.55 0.20 0.10 0.05 Fre ._ uency Which of the following statements are true? W 19. The followin table re I. The mean and modal number of accidents are equal. 11. The mean and median number of accidents are equal. III. The median and modal number of accidents are equal. A. I only B. II only C. HI only D. I, II, and III E. The correct answer is not given by A, B, C, or D 20. Let X,,Xz and X3 be independent, identically distributed random variables, each with density function fx(x)={ Let y =max{X,,X,,X,}. What is m[r >3? 3x2 forOSxSl, 0 otherwise. ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski - 581 - SECTION 21 1 37 (3 353 511 A. — . . —— 64 64 512 ' 512 €31 l—t for 21. Let the random variable X have the moment generating function M (t) = 2 —1 < t <1. Find the mean and variance of X. A.1and2 B.land3 C.3and2 D.3and3 E.3and6 22. A random sample of size 6 is selected with replacement from an urn that contains 10 red, 5 white, and 5 blue balls. What is the probability that the sample contains 2 balls of each color? .1 .1 Ci Di E12 1024 646 512 646 64 23. A drawer contains 6 blue socks and 4 white socks. Two of the socks are chosen are random and without replacement. What is the probability that the two socks are the same color? A. l B. l c. 13 D. 31 E. E 3 15 25 15 25 24. Let (X ,Y ) be distributed uniformly on the circular disk centered at (0,0) with radius %. 71: 2 2 What is the marginal density of X for —— _<. x S —? \EE ‘5? A.2 i-x2 Bi C. i—xz D.-1-‘F1-—x2 E. 141—)? 7t 4 2r 2 n: 4 7t 25. A certain game involved rolling a pair of dice and watching for “sevens” to occur. What is the probability that it takes exactly 10 rolls to observe 8 sevens? 352 52 52 352 45-52 A. 36—9- B. E C. E D. 3610 E. 6'0 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski - 582 - PRACTICE EXAM l7 3 It _ -x g/ 26. Let the distribution function of X for x > 0 be Fx (1:) = 1— Z x k7 . What is the density lino function of X for x > 0? A. e" B. 27. Let X and Yhave the joint density function x+y for0<x<1and0<y<1, fx.y(x.y)={0 otherwise. What is the conditional mean E (Y IX = «a? 2+6y B A. l . .l 5 3 12 2 28. Let X and Y have joint probability function w fowl”; F133, fxy (1‘9”: 36 0 otherwise. Which of the following statements is true? A. X and Y are dependent random variables with different marginal probability functions B. X and Y are dependent random variables with the same marginal probability function C. X and Y are independent random variables with different marginal probability functions D. X and Y are independent random variables with the same marginal probability function E. There is insufficient information to determine if X and Y are dependent or independent 29. Let X and Y be continuous random variables with joint density function 3(2—x-y) for0<x<2,0<y<2, andx+y<2, fxx (103’) = 4 0 otherwise. What is the conditional probability Pr(X <1|Y <1)? A 3.3 c.— 0.9 El 4 7 8 l ' 2 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski - 583 - SECTION 21 30. Let X have the density function 2_x fx (x) = k; 0 otherwise. forOS'xSk, For what value of k is the variance of X equal to 2? A.2 ~ B.6 C.9 D.18 E.36 ASM Study Manual for Course Pll Actuarial Examination. © Copyright 2004-2008 by Knysztof Ostaszewski -584- ...
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