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Probability And Mathematical Statistics (Exam 1/P) Question Paper

Probability And Mathematical Statistics (Exam 1/P)

Course:Actuarial Science (Insurance)

Exam Year:

Joint Exam 1/P Sample Exam 2
Take this practice exam under strict exam conditions: Set a timer for 3 hours; Do not stop the timer
for restroom breaks; Do not look at your notes. If you believe a question is defective or poorly worded,
you must continue on just like during the real exam.
Video solutions are available for this exam at http://www.theinniteactuary.com/?page=exams&id=50
TIA 1/P Seminar p. 1 Sample Exam 2
1. Find the mode of a Poisson random variable with mean 1.5.
A. 0 B. 0:5 C. 1 D. 1:5 E. 2
2. X and Y are identically distributed, independent random variables with cdf F(x) = 1??e??2x for x > 0.
What is P[3X + 2Y > 5]?
A. 0:1 B. 0:3 C. 0:5 D. 0:7 E. 0:9
3. Losses have a distribution whose density is 3y??4 for y > 1. If there is a deductible of 2, what is the
expected size of a payment.
A. 1=8 B. 2=8 C. 3=8 D. 4=8 E. 5=8
4. The density of X is given by f(x) =
2
(1 + 4x2)
for ??1 < x < 1. Find the density fY (y) of Y = 1=X
for y 6= 0.
A.
2y2
(y2 + 4)
B.
2
(y2 + 4)
C.
2
(1 + 4y2)
D.
2y2
(1 + 4y2)
E.
4
(1 + 4y2)
5. What is the smallest possible value of P[AB] if P[A] = 0:7 and P[B] = 0:8?
A. 0:4 B. 0:5 C. 0:6 D. 0:7 E. 0:8
6. X and Y have joint density f(x; y) = (1 + x3)??2 for 0 < y < 3x2 < 1. Find Var[Y j X = x]
A. x2
4
B. x4
4
C.
3x4
4
D. 9x4 E.
9x4
(1 + x3)2
7. The number of commercials C during the rst 200 miles of the Daytona 500 is a random variable with
distribution equal to 10 plus a Poisson random variable with mean 3.
What is the coecient of variation of C?
A. 0:13 B. 0:58 C. 1:00 D. 1:45 E. 1:73
8. In a blind tasting of wines from Bordeaux and Napa valley, a wine expert has a 90% chance of correctly
identifying that a wine is from Bordeaux and an 80% chance of correctly identifying a wine from Napa
valley. If 60% of the wines at the tasting are from Napa, what is the probability that a randomly
selected wine is from Napa valley, given that the wine expert said that it was from Napa valley?
A. 0:60 B. 0:80 C. 0:83 D. 0:90 E. 0:92
TIA 1/P Seminar p. 2 Sample Exam 2
9. A bag contains three dice, two of which are somewhat unusual: die A is a regular die whose faces show
1; 2; 3; 4; 5 and 6. Die B has a 4 on all 6 sides, and the faces of die C are 2; 2; 4; 4; 6 and 6. Suppose that
a die is randomly selected, rolled, and comes up a 4. If the same die is rerolled, what is the probability
that it is again a 4?
A. 0:50 B. 0:51 C. 0:59 D. 0:67 E. 0:76
10. A box contains 10 poorly stored lightbulbs, 4 of which are defective. If I randomly select 3 of the
lightbulbs to use in my living room, what is the probability that at least one is defective?
A. 0:72 B. 0:75 C. 0:78 D. 0:80 E. 0:83
11. If the moment generating function of X is e5et+2t??5, what is the variance of X?
A. 5 B. 7 C. 25 D. 49 E. 54
12. If P[A j B] = :05 + P[B j A], P[AB] = :1, and P[A] = :5, nd P[B].
A. 0:2 B. 0:3 C. 0:4 D. 0:5 E. 0:6
13. The number of chips per cookie in a batch of chocolate chip cookies has a Poisson distribution with
mean 5. Assuming that the number of chips per cookie is indepent, use a normal approximation with
a suitable continuity correction to estimate the probability that a dozen cookies contain fewer than 50
chocolate chips.
A. 0:02 B. 0:09 C. 0:19 D. 0:28 E. 0:35
14. The joint density of X and Y is
f(x; y) =
(
2
9 (x3 + y3) 0 < x < 1; 0 < y < 2
0 otherwise.
Find Emax[(X; Y )]
A. 0:1 B. 0:7 C. 1:0 D. 1:3 E. 1:6
15. In a shooting contest, a contestant has a probability of :7 of hitting the rst two targets, and a
probability of only :4 of hitting the third and fourth targets. If each shot is independent, what is the
probability that she hits at least three targets?
A. 0:08 B. 0:15 C. 0:23 D. 0:30 E. 0:38
16. For a random variable X, let g(X) be given by g(X) = EjX ?? m(X)j, where m(X) is the median of
X. Find g(X) for a Poisson random variable with mean 1.5.
A. 0:5 B. 0:7 C. 0:9 D. 1:0 E. 1:1
TIA 1/P Seminar p. 3 Sample Exam 2
17. The joint density of X and Y is 3??x??y for 0 < x < y < 1 and 0 otherwise. Find the expected value
of X.
A. 0:1 B. 0:3 C. 0:5 D. 0:7 E. 0:9
18. Suppose that the moment generating function of (X1;X2) is given byMX1;X2(t1; t2) = e2t1??3t2+t21
+2t22
??t1t2 .
Find the variance of X1.
A. 1 B. 2 C. 3 D. 4 E. 5
19. At a small college, the number of students taking calculus is 320. There are 295 students taking physics,
and 185 taking chemistry. The number taking calculus and physics is 140, and there are 105 taking
both physics and chemistry. If the number of students taking calculus and physics and chemistry is
60, and the number taking only chemistry is 80 less than the number taking only physics, how many
students are taking a class in at least one of these three subjects?
A. 445 B. 465 C. 485 D. 505 E. 525
20. The joint mgf of (X1;X2) is given by
MX1;X2 = e3et1+t22
??3t2??3:
Find E[2X1 ?? X2].
A. 1 B. 3 C. 5 D. 7 E. 9
21. A company insures 2 machines for maintainance. In a given year, each machine will require main-
tainance with probability 1/3, and if maintainance is required, the cost will be a uniform random
variable between $0 and $4,000. If the insurance policy has an annual payment limit of $6,000 for both
machines combined, what is the expected annual payment made by the insurer?
A. 1315 B. 1325 C. 1335 D. 1345 E. 1355
22. Let X be the outcome from rolling a fair six-sided die, and let N be the number of heads in X
independent
ips of a coin. If the die roll is independent of the coin tosses, then what is EN?
A. 1:5 B. 1:75 C. 2 D. 3 E. 3:5
23. If I roll 6 dierent fair six-sided dice, what is the probability that the maximum of the rolls is equal
to 5?
A. 0:09 B. 0:25 C. 0:33 D. 0:51 E. 0:83
TIA 1/P Seminar p. 4 Sample Exam 2
24. A company pays an annual insurance premium of $1,000 at the beginning of the year to insure their
main assembly line. If they have to pay the premium each year until there is a failure on the line,
what is the expected total premium paid if the probability that the line is still working after t years is
e??t=2?
A. $650 B. $930 C. $1540 D. $2000 E. $2540
25. The computer network for a company has two servers. If the failure times are independent and the
time until one server fails is uniformly distributed on [0; 20], and the other is uniformly distributed on
[0; 30], what is the variance of the time until at least one of the servers fails?
A. 16 B. 19 C. 22 D. 25 E. 28
26. An insurance policy reimburses a loss up to a benet limit of 20. If the distribution of a loss is
exponential with mean 30, what is the expected value of the benet paid under the policy?
A. 10 B. 15 C. 20 D. 25 E. 30
27. Suppose that the joint density of X and Y is (12=5)[x+y2] for 0 < x < 1 and 0 < y < x. What is the
covariance of X and Y ?
A. 0 B. :02 C. :12 D. :34 E. :66
28. Two fair six-sided dice are rolled. If the sum of the rolls is 10, what is the probability that exactly one
of the dice comes up a six?
A. 1=6 B. 2=6 C. 3=6 D. 4=6 E. 5=6
29. Losses X for a certain type of insurance have a lognormal distribution given by X = eZ, where Z is a
normal random variable with mean 2 and variance 3. The following year, losses are 10% higher due to
in
ation. What is the probability that a loss the next year exceeds 40?
A. :14 B. :18 C. :22 D. :26 E. :30
30. When I am sick, I like to count the buses that pass my house. I have noticed that the number of buses
that pass can be modelled quite well by a Poisson random variable with mean 8 per hour from 8 until
10 am, and by a Poisson random variable with mean 5 per hour from 10 am until 2 pm. If exactly 5
buses drive past my house from 9:30-10:30 one morning, what is the probability that exactly 11 buses
drove past from 9 to 11?
A. :02 B. :16 C. :26 D. :38 E. :50
TIA 1/P Seminar p. 5 Sample Exam 2
Answers
(1) C
(2) A
(3) A
(4) B
(5) B
(6) C
(7) A
(8) E
(9) E
(10) E
(11) A
(12) C
(13) B
(14) E
(15) E
(16) C
(17) B
(18) B
(19) D
(20) E
(21) B
(22) B
(23) B
(24) E
(25) E
(26) B
(27) B
(28) D
(29) B
(30) B
TIA 1/P Seminar p. 6 Sample Exam 2

Probability And Mathematical Statistics (Exam 1/P) Question Paper

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