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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(ZXY $5) in terms of (I)? ”(J—z.) we) c it?) we) we.) ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 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 chisquare 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)” Bere)” atlas1') ”14%)”. .0
E. [14%) W l ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 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 twostep 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 20042008 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(Ac) = 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 email to her professor teaching the class. One out of
every thousand emails 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 email 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 sixfaced 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 20042008 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 threemonth 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
threemonth 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 xyplane
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 20042008 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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