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Unformatted text preview: PRACTICE EXAlVIINATION NUMBER 2 1. Let X1}: ,X3 be a random sample from a discrete distribution with probability function 1, forx=0, 3
p(x)= %, forx=l,
0 otherwise. Determine the moment generating function M (t) of Y = XlX2X3. A.1—9+—8e' 1LT D183. E.12...
27 27 B.l+2e' C. —+—e .— 
(3 3 27 27 2. A doctor is studying the relationship between blood pressure and heartbeat abnormalities in
her patients. She tests a random sample of her patients and notes their blood pressures (high,
low, or normal) and their heartbeats (regular or irregular). She ﬁnds that: (i) 14% have high blood pressure. (ii) 22% have low blood pressure. (iii) 15% have an irregular heartbeat. (iv) Of those with an irregular heartbeat, onethird have high blood pressure. (v) Of those with normal blood pressure, oneeighth have an irregular heartbeat. What portion of the patients selected have a regular heartbeat and low blood pressure? A. 2%  B. 5% C. 8% D. 9% E. 20% 3. An actuary studying the 1nsurance preferences of automobile owners makes the following
conclusions:  (i) An automobile owner is twice as likely to purchase collision coverage as diSability coverage.
(ii) The event that an automobile owner purchases collision coverage is independent of the eVent
that he or she purchases disability coverage. 9 A (iii) The probability that an automobile owner purchases both collision and disability coverages
is 0.15. What 15 the probability that an automobile owner purchases neither collision nor disability
coverage? A. 0.18 B. 0.33 C. 0.48 D. 0.67 E. 0.82 ASM Study Manual for Course PI] Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski ‘ ‘ ' 1  141  PRACTICE EXAMINATION NO. 2
4. The monthly proﬁt of Company I can be modeled by a continuous random variable with density function f. Company 11 has a monthly proﬁt that is twice that of Company I. Determine
the probability density function of the monthly proﬁt of Company II. A. %f(§] 13.4325] 0211325) D. 2f(x) E.2f(2x) 5. The number of days that elapse between the beginning of a calendar year and the moment a
highrisk driver is involved in an accident is exponentially distributed. An insurance company
expects that 30% of highrisk drivers will be involved in an accident during the ﬁrst 50 days of a
calendar year. What portion of highrisk drivers are expected to be involved in an accident
during the ﬁrst 80 days of a calendar year? A. 0.15 B. 0.34 C. 0.43 D.0.57 E. 0.66 6. An insurance company insures a large number of drivers. Let X be the random variable
representing the company’s losses under collision insurance, and let Y represent the company’s
losses under liability insurance. X and Yhave joint density function 2x+2y
f(x.y)= 4 ’ 0, otherwise.
What is the probability that the total loss is at least 1? for0<x<land0<y<2, A. 0.33 B. 0.38 C. 0.41 D. 0.71 E. 0.75 7. The proﬁt for a new product is given by Z = 3X — Y — 5. X and Y are independent random
variables with Var(X) = l and Var(Y) = 2. What is the variance of Z? A. l B. 5 C. 7 D. 11 E. 16 8. A device contains two circuits. The second circuit is a backup for the ﬁrst, so the second is used only when the ﬁrst has failed. The device fails when and only when the second circuit fails.
Let X and Y be the times at which the ﬁrst and second circuits fail, respectively. X and Y have
joint probability density ﬁmction ' 6e‘xe'z’ , for 0 < x < y < oo,
f (M) = .
0, otherwrse.
What is the expected time at which the device fails? A. 0.33 B. 0.50 C. 0.67 D. 0.83 E. 1.50
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  142  PRACTICE EXAMINATION NO. 2 9. The probability that a visit to a primary care physician’s (PCP) ofﬁce results in neither lab
work nor referral to a specialist is 35%. Of those coming to a PCP’s ofﬁce, 30% are referred to
specialists and 40% require lab work. Determine the probability that a visit to a PCP’s ofﬁce
results in both lab work and referral to a specialist. A. 0.05 B. 0.12 C. 0.18 D. 0.25 E. 0.35 10. A study of automobile accidents uroduced the following data: An automobile from one of the model years 1997, 1998, and 1999 was involved in an accident.
Determine the probability that the model year of this automobile is 1997. A. 0.22 B. 0.30 C. 0.33 D. 0.45 E. 0.50 11. The lifetime of a printer costing 200 is exponentially distributed with mean 2 years. The
manufacturer agrees to pay a full refund to a buyer if the printer fails during the ﬁrst year
following its purchase, and a onehalf refund if it fails during the second year. If the
manufacturer sells 100 printers, how much should it cxpectto pay in refunds? A. 6,321 B. 7,358 C. 7,869 D;'10,256 E. 12,642 12. Let T denote the time in minutes for a customer service representative to respond to 10
telephone inquiries. T is uniformly distributed on the interval with endpoints 8 minutes and 12
minutes. Let R denote the average rate, in customers per minute, at which the representative
responds to inquiries. Which of the following is the density function of the random variable R on the interval [19 .19]?
12 8 AE B. 3——5— c. 3r—51”(r) D. § E. i,
5 2r 2 r 2r 13. Let TI and T2 represent the lifetimes in hours of two linked components in an electronic ASM Study Manual for Course P/l Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  143  PRACTICE EXAMINATION NO. 2
device. The joint density function for T, and T2 is uniform over the region deﬁned by 0 S tl S 1,. S L where L is a positive constant. Determine the expected value of the sum of the
squares of Tiand T2. 14. Two instruments are used to measure the height, h, of a tower. The error made by the less
accurate insn'ument is normally distributed with mean 0 and standard deviation 0.0056h. The
error made by the more accurate instrument is normally distributed with mean 0 and standard
deviation 0.004411. Assuming the two measurements are independent random variables, what is
the probability that their average value is within 0.005h of the height of the tower? A. 0.38 B. 0.47 C. 0.68 D. 0.84 E. 0.90 15.‘An insurance company’s monthly claims are modeled by a continuous, positive random
variable X, whose probability density function is proportional to (1+ x)‘4 , where 0 < x < co.
Determine the company’s expected monthly claims. D.l E.3 A B l 1 l
.— .— C.—
6 3 2 16. A probability distribution of the claim sizes for an auto insurance policy is given in the table below:
Size
50 0.20
0.10
0.10
0.30
What percentage of the claims are within one standard deviation of the mean claim size? 70 A. 45% B. 55% C. 68% D. 85% E. 100% 17. The total claim amount for a health insurance policy follows a distribution with density ASM Study Manual for Course PI] Actuarial Examination. © Copyright 20042008 by Krzysztof Osmszewski  144  J PRACTICE EXAMINATION NO. 2 function f (x) = ﬁem for x > 0.The premium for the policy is set at 100 over the expected total claim amount. If 100 policies are sold, what is the approximate probability that the
insurance company will have claims exceeding the premiums collected? A. 0.001 B. 0.159 C. 0.333 D. 0.407 E. 0.460 18. An insurance company sells two types of auto insurance policies: Basic and Deluxe. The
time until the next Basic Policy claim is an exponential random variable with mean two days.
The time until the next Deluxe Policy claim is an independent exponential random variable with
mean three days. What is the probability that the next claim will be a Deluxe Policy claim? A. 0.172 B. 0.223 C. 0.400 D. 0.487 E. 0.500 19. A company offers a basic life insurance policy to its employees, as well as a supplemental
life insurance policy. To purchase the supplemental policy, and employee must ﬁrst purchase
the basic policy. Let X denote the proportion of employees who purchase the basic policy, and Y
the proportion of employees who purchase the supplemental policy. Let X and Y have the joint density function f (x, y) = 2(x + y) on the region where the density is positive. Given that 10% of the employees buy the basic policy, what is the probability that fewer than 5% buy the
supplemental policy? A. 0.010 B. 0.013 C. 0.108 D. 0.417 E. 0.500 20. An insurance policy reimburses dental expense, X, up to a maximum beneﬁt of 250. The
probability density function for X is: f(x) ce'°”°"‘, forx20,
0, otherwise,
where c is a constant. Calculate the median beneﬁt for this policy. A. 161 B. 165 C. 173 D. 182 E. 250 21. In an analysis of healthcare data, ages have been rounded to the nearest multiple of 5 years.
The difference between the true age and the rounded age is assumed to be uniformly distributed
on the interval from ~25 years to 2.5 years. The healthcare data are based on a random sample
of 48 people. What is the approximate probability that the mean of the rounded ages is within
0.25 years of the mean of the true ages? A A. 0.14 B. 0.38 C. 0.57 D. 0.77 E. 0.88 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  145  PRACTICE EXAMINATION NO. 2
22. Let X and Y denote the values of two stocks at the end of a fiveyear period. X is uniformly distributed on the interval (0,12). Given X = x, Y is uniformly distributed on the interval
(0,x) . Determine Cov(X,Y) according to this model. A.0 B.4 C.6 D. 12 E. 24 23. An actuary determines that the annual numbers of tornadoes in counties P and Q are jointly
distributed as follows: Annual number of Tornadoes in county Q 0 1 2 Annual number . . . 0.02
of tornadoes 1 0.13 0.15 0.12 0.03
in county P 2 0.05 0.15 0.10 0.02 Calculate the conditional variance of the annual number of tornadoes in county Q, given that
there are no tornadoes in county P. A.‘0.51 B. 0.84 C. 0.88 D. 0.99 E. 1.76 '7 .124. An insurance policy is written to cover a loss X where X has density function 2x2, forOSxSZ, f (x)= 0, otherwise. The time (in hours) to process a claim of size x, where 0 < x < 2, is uniformly distributed on the 7 interval from x to 2x. Calculate the probability that a randomly chosen claim On this pohcy lS
processed 1n three hours or more. ~ A. 0.17 B. 0.25 c. 0.32 D. 0.53 E. 0.83  T 25. An actuary has discovered that policyholders are three times as likely to ﬁle th claims as to .  ﬁle four claims. If the number of claims ﬁled has a Poisson distribution, what is the variance of
the number of claims ﬁled? a A.—1— 13.1 c../2 13.2 B.4 J3 26. A car dealership sells 0, 1, or 2 luxury cars on any day. When selling a car, the dealer also ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  146  PRACTICE EXAMINATION N0. 2
tries to persuade the customer to buy an extended warranty for the car. Let X denote the number of luxury cars sold in a given day, and let Y denote the number of extended warranties sold. pr(x=o,y=o)=l, Pr(X=1,Y=0)=—1—, Pr(X=2,Y=0)=—1,
6 12 12
Pr(X=l,Y=1)=;, m(x=2,r=1)=;, P(X=2,Y=2)=—;.
What is the variance of X?
A. 0.47 B. 0.58 c. 0.83 D. 1.42 E. 258 27. A blood test indicates the presence of a particular disease 95% of the time when the disease
is actually present. The same test indicates the presence of the disease 0.5% of the time when the
disease is not present. One percent of the population actually has the disease. Calculate the
probability that a person has the disease given that the test indicates the presence of the disease. A. 0.324 B. 0.657 C. 0.945 D. 0.950 E. 0.995 28. An insurance policy reimburses a loss up to a beneﬁt limit of 10. The policyholder’s loss, Y,
follows a distribution with density function: 1
f(y)= y” 0, otherwise.
What is the expected value of the beneﬁt paid under the insurance policy? fory>1, A. 1.0 B. 1.3 C. 1.8 D. 1.9 E. 2.0 29. A company insures homes in three cities, J , K, and L. Since sufﬁcient distance separates the
cities, it is reasonable to assume that the losses occurring in these cities are independent. The moment generating functions for the loss distributions of the cities are: M J (t) = (l — 2t)‘3 ,
M K (t) = (l  2025 , M ,_ (t) = (1 — 20—45 . Let X represent the combined losses from the three cities. Calculate E (X 3). A. 1,320 B. 2,082 C. 5,760 D. 8,000 B. 10,560 30. In modeling the number of claims ﬁled by an individual under an automobile policy during a
threeyear period, an actuary makes the simplifying assumption that for all integers n 2 0, ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  147  PRACTICE EXAMINATION NO. 2 ﬂ p"+1 = % p,l , where p" represents the probability that the policyholder ﬁles 11 claims during the J period. Under this assumption, what is the probability that a policyholder ﬁles more than one
claim during the period? A.0.04 B.0.16 C.0.20 D.0.80 E.0.96 148 J ASM Study Manual for Course Pll Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski ...
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