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Unformatted text preview: PRACTICE EXAMINATION NUMBER 1 1. A company agrees to accept the highest of four sealed bids on a property. The four bids are
regarded as four independent random variables with common cumulative distribution function F(x) = i0 + sinn'x) 3 5 . . .
for 5 S x S Which of the followmg represents the expected value of the accepted bid? A. 7: xcosn’x dx B (1+sin7rx)4dx C. Ila x(l+sin7cx)4dx L
'16 nluI—.uu
talent—gulc
le'—.Mu l
D. 271' cosn’x(l+sin7rx)3dx E. 211—75 xcoszrx(l+sin7rx)3dx Mum—“om.
«clung—“91:1. 2. A company is reviewing tornado damage claims under a farm insurance policy. Let X be the
portion of a claim representing damage to the house and let Ybe the portion of the same claim
representing damage to the rest of the property. The joint density function of X and Y is 61 x+ forx>0, >0,x+ <1,
f(x,y)= ( ( y» . y y
0, otherw13e. Determine the probability that the portion of a claim representing damage to the house is less
than 0.2. A. 0.360 B. 0.480 C. 0.488 D. 0.512 E. 0.520 3. An insurance company issues life insurance policies in three separate categories: standard,
preferred, and ultrapreferred. 0f the company’s policyholders, 50% are standard, 40% are
preferred, and 10% are ultrapreferred. Each standard policyholder has probability 0.010 of
dying in the next year, each preferred policyholder has probability 0.005 of dying in the next
year, and each ultrapreferred policyholder has probability 0.001 of dying in the next year. A
policyholder dies in the next year. What is the probability that the deceased policyholder was
ultra—preferred? A.0.0001 B.0.0010 C.0.0071 D.0.0141 E.0.2817 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  110  PRACTICE EXAMINATION N0. 1
4. A joint density function is given by
kx, for0<x<l, 0<y<1, f(x’y) = {0, otherwise,
where k is a constant. What is Cov(X,Y )? A. 1 3.0 C .l D
6 9 l
' 6 5. An actuary is studying the prevalence of three health risk factors, denoted by A, B, and C,
within a population of women. For each of the three factors, the probability is 0.1 that a woman
in the population has only this risk factor (and no others). For any two of the three factors, the
probability is 0.12 that she has exactly these two risk factors (but not the other). The probability that a woman has al three risk factors, given that she has A and B, is What is the probability that a woman has none of the three risk factors, given that she does not have risk factor A? A. 0.280 B. 0.311 C. 0.467 D. 0.484 E. 0.700 6. Two life insurance policies, each with a death beneﬁt of 10,000 and a onetime premiumdof
500, are sold to a couple, one for each person. The policies will expire at the end of the tenth
year. The probability that only the wife will survive at least ten years is 0.025, the probability
that only the husband will survive at least ten years is 0.01 , and the probability that both of them
will survive at least ten years is 0.96. What is the expected excess of premiums over claims,
given that the husband survives at least ten years? A. 350 B. 385 C. 397 D. 870 E. 897 7. You are given Pr(AuB)=0.7 and Pr(AUBC)=0.9. Determine Pr(A). A. 0.2 B. 0.3 C. 0.4 D. 0.6 D. 0.8 8. A study is being conducted in which the health of two independent groups of ten policyholders
is being monitored over a oneyear period of time. Individual participants in the study drop out
before the end of the study with probability 0.2 (independently of the other participants). What is
the probability that at least 9 participants complete the study in one of the two groups, but not in
both groups? A. 0.096 B. 0.192 C. 0.235 D. 0.376 E. 0.469
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  lll  PRACTICE EXAMINATION NO. 1
9. The stock prices of two companies at the end of any given year are modeled with random variables X and Y that follow a distribution with joint density function
2x, for0<x<1,x<y<x+1, fora) = {0, otherwise.
What is the conditional variance of 1' given that X = x? A.— B.1 C.x+1 D. chl E.x2+x+l
12 6 2 6 3 10. An auto insurance company insures an automobile worth 15000 for one year under a policy
with a 1000 deductible. During the policy year there is a 0.04 chance of partial damage to the
car and a 0.02 chance of a total loss of the car. If there is partial damage to the car, the amount X
of damage (in thousands) follows a distribution with a density function f(x) = {0.500% 2, for 0 <x <15, 0, otherwise.
What is the expected claim payment? A. 320 B. 328 C. 352 D. 380 E. 540 11. A company manufactures a brand of light bulb with a lifetime in months that is normally
distributed with mean 3 and variance 1. A consumer buys a number of these bulbs with the
intention of replacing them successively as they burn out. The light bulbs have independent
lifetimes. What is the smallest number of bulbs to be purchased so that the succession of light
bulbs produces light for at least 40 months with probability at least 0.9772? A. 14 B. 16 C. 20 D40 D.55 12. A device that continuously measures and records seismic activity is placed in a remote
region. The time, T, to failure of this device is exponentially distributed with mean 3 years.
Since the device will not be monitored during its ﬁrst two years of service, the time to discovery of its failure is X = max(T,2). Determine 2 4 2
1 A. 2+3e'6 B. 22e’3 +563 c. 3 D. 2+3e"3 E. 5 13. The waiting time for the ﬁrst claim from a good driver and the waiting time for the ﬁrst claim
from a bad driver are independent and follow exponential distributions with means 6 years and 3
ASM Study Manual for Course Pl] Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  112  PRACTICE EXAMINATION NO. 1
years, respectively. What is the probability that the ﬁrst claim from a good driver will be ﬁled within 3 years and the ﬁrst claim from a bad driver will be ﬁled within 2 years? 1 2 1 1 1 1 2 1 1
A.—1e3e2+e6 B.—e‘ C.1—e3—e2+e6
l8 l8
2 1 1 12 1 1 1
D.l—e3—e2+e3 E.1—e3e2i—e6
3 6 18 14. A hospital receives 20% of its ﬂu vaccine shipments from Company X and the remainder of
its shipments from other companies. Each shipment contains a very large number of vaccine
vials. For Company X’s shipments, 10% of the vials are ineffective. For every other company,
2% of the vials are ineffective. The hospital tests 30 randomly selected vials from a shipment
and ﬁnds that one vial is ineffective. What is the probability that this shipment came from
Company X? A. 0.10 B. 0.14 C. 0.37 D. 0.63 E. 0.86 15. A device contains two components. The device fails if either component fails. The joint
density function of the lifetimes of the components, measured in hours, is f (SJ), where 0 < s < 1, and 0 < t < 1.What is the probability that the device fails during the ﬁrst half hour of
operation? 05 ll A. ij(s,t)ds dt B. jjf(s,t)ds dt C. ij(s,t)ds dt
D. Tiﬂsﬂdr dt+ij(s,t)ds dt E. ij(s,t)ds dt + ij(s,t)ds dt 16. A company offers earthquake insurance. Annual premiums are modeled by an exponential
random variable with mean 2. Annual claims are modeled by an exponential random variable
with mean 1. Premiums and claims are independent. Let X denote the ratio of claims to
premiums. What is the density function of X? 1 B. —2— C. e" D. 2e'2’ E. xe" A.
2x+l (2x+ l)2 17. Claim amounts for wind damage to insured homes are independent random variables with ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  113  PRACTICE EXAMINATION N0. 1
common density function i forx>l. f(x)= 0, otherwise, where x is the amount of a claim in thousands. Suppose 3 such claims will be made. What is the
expected value of the largest of the three claims? A. 2025 B. 2700 C. 3232 D. 3375 E. 4500 18. A large pool of adults earning their ﬁrst driver’s license includes 50% lowrisk drivers, 30%
moderaterisk drivers, and 20% highrisk drivers. Because these drivers have no prior driving
record, an insurance company considers each driver to be randomly selected from the pool. This
month, the insurance company writes 4 new policies for adults earning their ﬁrst driver’s license.
What is the probability that these 4 will contain at least two more highrisk drivers than lowrisk
drivers? A.0.006 3.0.012 C.0.018 D.0.049 E. 0.0073 19. An insurer offers a health plan to the employees of a large company. As part of this plan, the
individual employees may choose exactly two of the supplementary coverages A, B , and C, or
they may choose no supplementary coverage. The proportions of the company’s employees that choose coverages A, B, and C are i, g, and %, respectively. Determine the probability that a randomly chosen employee will choose no supplementary coverage. A.0 Rﬂ C .l D.
144 2 144 20. A company has two electric generators. The time until failure for each generator follows an
exponential distribution with mean 10. The company will begin using the second generator
immediately after the ﬁrst one fails. What is the variance of the total time that the generators
produce electricity? A. 10 B. 20 C. 50 D. 100 E. 200 21. For Company A there is a 60% chance that no claim is made during the coming year. If one
or more claims are made, the total claim amount is normally distributed with mean 10000 and
standard deviation 2000. For Company B there is a 70% chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  114  PRACTICE EXAMINATION NO. I
with mean 9000 and standard deviation 2000. Assume that the total claim amounts of the two
companies are independent. What is the probability that, in the coming year Company B’s total
claim amount will exceed Company A’s total claim amount? A. 0.180 B. 0.185 C.0.217 D.0.223 E. 0.240 22. The warranty on a machine speciﬁes that it will be replaced at failure or age 4, whichever
occurs ﬁrst. The machine’s age at failure, X, has density function 1 for0<x<5. f(x)= 5’ 0, otherwise.
Let Y be the age of the machine at the time of replacement. Determine the variance of Y. A. 1.3 B. 1.4 C. 1.7 D. 2.1 E. 7.5 23. A tour operator has a bus that can accommodate 20 tourists. The operator knows that tourists
may not show up, so he sells 21 tickets. The probability that an individual tourist will not show
up is 0.02, independent of all other tourists. Each ticket costs 50, and is nonrefundable if a
tourist fails to show up. If a tourist shows up and a seat is not available, the tour operator has to
pay 100 (ticket cost + 50 penalty) to the tourist. What is the expected revenue of the tour operator? A. 935 B. 950 C. 967 D. 976 E. 985 24. An insurance company insures a large number of homes. The insured value, X, of a
randomly selected home is assumed to follow a distribution with density function 31:", forx>l,
x =
fX( ) {0, otherwise. Given that a randomly selected home is insured for at least 1.5, what is the probability that it is
insured for less than 2? A. 0.578 B. 0.684 C. 0.704 D. 0.829 E. 0.875 25. A public health researcher examines the medical records of a group of 937 men who died in
1999 and discovers that 210 of the men died from causes related to heart disease. Moreover, 312
of the 937 men had at least one parent who suffered from heart disease, and, of these 312 men,
102 died from causes related to heart disease. Determine the probability that a man randomly
selected from this group died of causes related to heart disease, given that neither of his parents ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Knysztof Ostaszewski  115  PRACTICE EXAMINATION NO. I
suffered from heart disease. A. 0.115 B.0.173 C.0.224 D.0.327 E.0.514 26. A recent study indicates that the annual cost of maintaining and repairing a car in a town in
Ontario averages 200 with a variance of 260. If a tax of 20% is introduced on all items associated
with the maintenance and repair of cars (i .e., everything is made 20% more expensive), what will
be the variance of the annual cost of maintaining and repairing a car? A. 208 B. 260 C. 270 D. 312 E. 374 27. An auto insurance company has 10,000 policyholders. Each policyholder is classified as (i) young or old; (ii) male or female; and (iii) married or single. Of these policyholders, 3000 are young, 4600 are male, and 7000 are married. The policyholders
can also be classiﬁed as 1320 young males, 3010 married males, and 1400 young married
persons. Finally, 600 of the policyholders are young married males. How many of the company’s
policyholders are young, female, and single? A. 280 B. 423 C. 486 D. 880 E. 896 28. A diagnostic test for the presence of a disease has two possible outcomes: 1 for disease
present and 0 for disease not present. Let X denote the disease state of a patient, and let Y denote
the outcome of the diagnostic test. The joint probability function of X and Y is given by: Pr(X = 0, Y =0): 0.800,
Pr(X =1, Y = 0)=0.050,
Pr(X = 0, Y =1)=0.025,
Pr(X= 1, Y =1)=0.125.
Calculate Var(YX =1). A. 0.13 B.0.15 C.0.20 D.0.51 E.0.71 29. An insurance company issues 1250 vision care insurance policies. The number of claims
ﬁled by a policyholder under a vision care insurance policy during one year is a Poisson random
variable with mean 2. Assume the numbers of claims ﬁeld by distinct policyholders are
independent of one another. What is the approximate probability that there is a total of between 2450 and 2600 claims during a oneyear period?
ASM Study Manual for Course P/l Actuarial Examination. (0 Copyright 20042008 by Krzysztof Ostaszewski  116  J PRACTICE EXAMINATION NO. 1
l A. 0.68 B. 0.82 C. 0.87 D. 0.95 E. 1.00 30. A group insurance policy covers the medical claims of the employees of a small company.
The value, V, of the claims made in one year is described by V =100000Y,where Y is a random variable with density function k 1— 4, for0< <1,
my)= ‘ y) ?’
0, otherwrse, where k is a constant. What is the conditional probability that V exceeds 40000, given that V
exceeds 10000? A. 0.08 B. 0.13 C. 0.17 D. 0.20 E. 0.51 ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 2004—2008 by Krzysztof Ostaszewski  117  ...
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This note was uploaded on 10/27/2010 for the course PSTAT 172a taught by Professor Staff during the Winter '08 term at UCSB.
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