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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 95th 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 20042008 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 20042008 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 szx—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 20042008 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 20042008 by Krzysztof Ostaszewski  175  PRACTICE EXAMINATION N O. 3
18. Let X and Y be random variables whose joint distribution is uniform over the halfdisc: {(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 ith 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 35—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 20042008 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 JEx, for0<x<\/§,
f(x)= .
0, otherwrse.
What is the probability that exactly 2 of the 3 random variables exceed 1? A. gJi B. 3—2J5 c. 3(~/§1)(2—~/5)2
D(Eﬁllfél E3liJ5Tlﬁél 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 20042008 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 20042008 by Krzysztof Ostaszewski  178  .. J ...
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