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Unformatted text preview: PRACTICE EXAMINATION NUMBER 4 1. An insurance company sells an auto insurance policy that covers losses incurred by a
policyholder, subject to a deductible of 100. Losses incurred follow an exponential distribution
with mean 300. What is the 95th percentile of actual losses that exceed the deductible? A. 600 B. 700 C. 800 D. 900 E. 1000 2. An actuary determines that the claim size for a certain class of accidents is a random variable, X, with moment generating function M x (t) = W. Determine the standard deviation of
 t the claim size for this class of accidents. A. 1,340 B. 5,000 C. 8,660 D. 10,000 E. 11,180 3. An actuary studied the likelihood that different types of drivers would be involved in at least
one collision during any oneyear period. The results of the study are presented below. Type of driver Percentage Probability
of all drivers of at least
one collision Mil—_—
_—_ Given that a driver has been involved in at least one collision in the past year, what is the
probability that the driver is a young adult driver? A. 0.06 B. 0.16 C.0.19 D. 0.22 E. 0.25 4. A piece of equipment is being insured against early failure. The time from purchase until
failure of the equipment is exponentially distributed with mean 10 years. The insurance will pay
an amount x if the equipment fails during the ﬁrst year, and it will pay 0.5x if failure occurs
during the second or third year. If failure occurs after the ﬁrst three years, no payment will be made. At what level must x be set if the expected payment made under this insurance is to be
1000? A. 3858 B. 4449 C. 5382 D. 5644 E. 7235 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Knysztof Ostaszewski  199 ~ SECTION 8
5. A stock market analyst has recorded the daily sales revenue for two companies over the last
year and displayed them in the histograms below. Company A Company B
l104 ‘0 c... m
.3. £3?
E g 3%
Z 8 Z 8
a as??? e a WEE 3
Daily sales revenue Daily sales revenue The analyst noticed that a daily sales revenue above 100 for Company A was always
accompanied by a daily sales revenue below 100 for Company B, and vice versa. Let X denote
the daily sales revenue for Company A and let Y denote the daily sales revenue for Company B, on some future day. Assuming that for each company the daily sales revenues are independent
and identically distributed, which of the following is true? A. Var(X)>Var(Y) and Var(X+Y)>Var(X)+V
B. Var( )>Var(Y) andVar(X+Y)<Var(X )+Var( )
C Var(X )>Var(Y)andVar(X+Y)= Var(X)+Var(Y) ) ) ( )) a
A
5 D. Var(X <Var(Y )and Var(X+Y)>Var(X +Var Y
E. Var(X)<Var(Y) and Var(X+Y)<Var(X )+Var(Y 6. Claims ﬁled under auto insurance policies follow a normal distribution with mean 19,400 and
standard deviation 5,000. What is the probability that the average of 25 randomly selected claims
exceeds 20,000? A. 0.01 B. 0.15 C. 0.27 D. 0.33 E. 0.45 7. The future lifetimes (in months) of two components of a machine have the following joint
density function: 6
f“ (x.y) = 125,000
0: otherwise.
What is the probability that both components are still functioning 20 months from now? (SO—x—y), for0<x<50—y<50, ASM Study Manual for Course Pl] Actuarial Examination. © Copyright 20044008 by Knysztof Ostaszewski  200  PRACTICE EXAMINATION NO. 4 6 20 20 M) 50:
  dx 50—  dx
A 125,000! {(50 x y)dy 125 60002]; if x y)dy
6 30 SOxoy 50 50:
. so — dx so — dx
C 125,000,]; J, ( x y)!” 11.125000!0 if x my
6 50 50'1“!
50—  dx
125,000,!o Jo ( x my 8. The probability that a randomly chosen male has a circulation problem is 0.25. Males who
have a circulation problem are twice as likely to be smokers as those who do not have a
circulation problem. What is the conditional probability that a male has a circulation problem, given that he is a smoker? A. B. C. E D.
5 F11
who 1 l l
4 3 2 9. A company buys a policy to insure its revenue in the event of major snowstorms that shut
down business. The policy pays nothing for the ﬁrst such snowstorm of the year and 10,000 for
each one thereafter, until the end of the year. The number of major snowstorms per year that
shut down business is assumed to have a Poisson distribution with mean 1.5. What is the
expected amount paid to the company under this policy during a oneyear period? A. 2,769 B. 5,000 C. 7,231 D. 8,347 B. 10,578 10. A manufacturer’s annual losses follow a distribution with density function
2.5 0625 fx(x)= x” ’
0, otherwise. To cover its losses, the manufacturer purchases an insurance policy with an annual deductible of
2. What is the mean of the manufacturer’s annual losses not paid by the insurance policy? forx> 0.6, A. 0.84 B. 0.88 C. 0.93 D. 0.95 E. 1.00 11. An insurance policy is written to cover a loss, X, where X has a uniform distribution on
[0,1000]. At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible? ASM Study Manual for Course Pll Actuarial Examination. (9 Copyright 20042008 by Krzysztof Ostaszewski  201  SECTION 8
A. 250 B. 375 C. 500 D. 625 E. 750 12. A company prices its hurricane insurance using the following assumptions: (i) In any calendar year, there can be at most one hurricane. (ii) In any calendar year, the probability of a hurricane is 0.05. (iii) The number of hurricanes in any calendar year is independent of the number of hurricanes in
any other calendar year. Using the company’s assumptions, calculate the probability that there are fewer than 3 hurricanes
in a 20year period. A.0.06 B.O.l9 C.0.38 D.0.62 E.0.92 13. A marketing survey indicates that 60% of the population owns an automobile, 30% owns a
house, and 20% owns both an automobile and a house. Calculate the probability that a person
chosen at random owns an automobile or a house, but not both. A. 0.4 B.0.5 C.0.6 D.0.7 E. 0.9 14. Ten percent of a company's life insurance policyholders are smokers. The rest are
nonsmokers. For each nonsmoker, the probability of dying during the year is 0.01. For each
smoker, the probability of dying during the year is 0.05 . Given that a policyholder has died, what
is the probability that the policyholder was a smoker? A. 0.05 B. 0.20 C. 0.36 D. 0.56 E. 0.90 15. Let X and Ybe random losses with joint density function f“, (x, y) = e‘my) for x > 0 and y > 0. An insurance policy is written to reimburse X + Y. Calculate the probability that the
reimbursement is less than 1. A. e’2 B. e"1 C. l—e'l D. 1—2e'1 E. l—2e'2 16. As part of the underwriting process for insurance, each prospective policyholder is tested for
high blood pressure. Let X represent the number of tests completed when the ﬁrst person with
high blood pressure is found. The expected value of X is 12.5. Calculate the probability that the
sixth person tested is the ﬁrst one with high blood pressure. A. 0.000 B. 0.053 C. 0.080 D. 0.316 E. 0.394 ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  202  PRACTICE EXAMINATION NO. 4
17. The distribution of loss due to ﬁre damage to a warehouse is: _—
IRE—
—EEE_
—nm_ Given that a loss is greater than zero, calculate the expected amount of the loss. A. 290 B. 322 C. 1,704 D. 2,900 B. 32,222 18. An insurance policy covers the two employees of ABC Company. The policy will reimburse
ABC for no more than one loss per employee in a year. It reimburses the full amount of the loss
up to an annual companywide maximum of 8000. The probability of an employee incurring a
loss in a year is 40%. The probability that an employee incurs a loss is independent of the other
employee’s losses. The amount of each loss is uniformly distributed on [1000, 5000]. Given that
one of the employees has incurred a loss in excess of 2000, determine the probability that losses will exceed reimbursements. A.— B.— C.— D E l l
20 15 10 ' 8 ' 6 19. Let X and Y be discrete random variables with joint probability function
2“” , for (x,y)=(0,l),(0,2)(1,2),(1,3), P(x,y) =
0, otherwise.
Determine the marginal probability function of X. l
—, forx=0, —l, forx=0, 1, forx=0,
6 4 3
5 3 2
A. p(x)= g, forx=l, B.p(x)= Z, forx=1, C. p(x)= 3, forx=1,
0, otherwise. 0, otherwise. 0, otherwise.
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 2004—2008 by Knysztof Ostaszewski  203  SECTION 8 2
'9" forxﬁ’ y forx=0
3 12’ ’
—, forx=2, 2+y D. p(x)= 9 E.p(x)= —1§—, forx=1,
4
3. forx = 3. 0, otherwise.
0, otherwise. 20. Workplace accidents are categorized into three groups: minor, moderate and severe. The
probability that a given accident is minor is 0.5, that it is moderate is 0.4, and that it is severe is
0.1 . Two accidents occur independently in one month. Calculate the probability that neither
accident is severe and at most one is moderate. A. 0.25 B. 0.40 C. 0.45 D. 0.56 E. 0.65 21. A box contains 35 gems, of which 10 are real diamonds and 25 are fake diamonds. Gems are
randomly taken out of the box, one at a time without replacement. What is the probability that
exactly 2 fakes are selected before the second real diamond is selected? — — [Tl—J] minister Bldg—KEY 4 22. Let X and Ybe continuous random variables with joint density function 1
—e"', for —y<x<y andy>0,
fo (x,y) = 2
0, otherwise.
What is Pr(X < lY = 3)?
1 l 2
A. le‘3 B. 2e"3 C. —e’1 —e’3 D. — E. —
2 2 6 3
23. Let X be a continuous random variable with density function
e", forx > 0,
x =
fX( ) {0, . otherwise.
ASM Study Manual for Course P/l Actuarial Examination. © Copyright 20042008 by Krzysztof Ostaszewski  204 ~ PRACTICE EXAMINATION NO. 4
and let Y = X 2 — 1. What is the value of the cumulative distribution function of Y at y = 3? A. ge'a B. 'jIe'z C. l—2e’2 D. l—e‘2 E. l—e'3 24. Let X be a random variable with a uniform distribution on the interval (1, a) where a > 1. If
E(X)= 6 Var(X), then a = A.2 B.3 C.3J§ D.7 E.8 25. A hat contains 3 chips numbered 1, 2, and 3. Two chips are drawn successively from the hat
without replacement. What is the correlation between the number on the first chip and the
number on the second chip? A.—— B. — C.0 D E 1 l
2 3 ' 3 '2 26. Let X, Y, and Z have means 1, 2, and 3, respectively, and variances 4, 5, and 9, respectively.
The covariance of X and Y is 2, the covariance of X and Z is 3, and the covariance of Y and Z is 1.
What are the mean and variance, respectively, of the random variable 3X + 2Y — Z? A.4and31 B.4and65 C.4and67 D. l4and 13 E.l4and65 27. Let X be a continuous random variable with density function 3 2 2 1
0x+ 02x, for0<x<— fx (x) = E J5 ’
0, otherwise,
where 9 > 0. What is the expected value of X? 5 7
95 395 5 5 17
A.——+u—— 3.1 c —— D.— E.———
3 s 2J5 2 24¢? 28. Let X and Y be continuous random variables with joint density function
2, for0<x<y<l,
fxx (x ’y ) = { 0, otherwise.
ASM Study Manual for Course [’11 Actuarial Examination. © Copyright 20042008 by Knyszlof Ostaszewski  205  SECTION 8
What is the joint density function of U = XY and W = X, and where is it nonzero? A.£forw2<u<w<l B. 8uwfor0<u<w<l C.£foru2<w<u<1
w w
D.2forw2<u<w<l E.2foru2<w<u<l 29. A system made up of 7 components with independent, identically distributed lifetimes will
operate only until any one of the system’s components fails. If the lifetime X of each component
has density function i forx>1 fx (x)= x4 0, otherwise,
what is the expected time until failure of the system? A. 1.02 B. 1.03 C. 1.04 D. 1.05 E. 1.06 30. The number of power surges in an electric grid has a Poisson distribution with a mean of 1
power surge every 12 hours. What is the probability that there will be no more than 1 power
surge in a 24hour period? A. 2e‘2 B. 3e’2 C. e 2 D. N
MI— E. 3e" NIL» ASM Study Manual for Course P/l Actuarial Examination. ((3) Copyright 20042008 by Knysztof Ostaszewski  206  ...
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