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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 4
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 4
1. The cdf of the number of hours it takes a consultant to complete a project is given by F(x) = x2=16
for 0 x 4. The consultant bills $300 per hour, rounded up to the nearest half hour, for the project.
What is the expected amount of the total bill?
A. 722 B. 800 C. 872 D. 963 E. 1200
2. X and Y are uniformly chosen over pairs of integers with 1 X Y 3. Find P[X + Y 4].
A.
1
6
B.
2
6
C.
3
6
D.
4
6
E.
5
6
3. Suppose that X is a Poisson random variable with mean 2:5 and Y is a geometric random variable on
1; 2 : : : with mean 2:5. Let P denote the probability that X is equal to its mode, and let Q denote the
probability that Y is equal to its mode. Find jP ?? Qj.
A. 0:03 B. 0:14 C. 0:19 D. 0:26 E. 0:40
4. If X is an exponential random variable with mean 2 and Y is an independent exponential random
variable with mean 3, what is P[X < Y ]?
A. 0:3 B. 0:4 C. 0:5 D. 0:6 E. 0:7
5. An actuary for an insurance company models annual claims with a lognormal random variable X. If
ElnX = 1:08 and Var lnX = :25, what is P[X < 1]?
A. 0:02 B. 0:14 C. 0:44 D. 0:56 E. 0:98
6. Suppose that X is an exponential random variable with mean 2 and Y = j2X ?? 3j. Find fY (1).
A. 0:02 B. 0:09 C. 0:15 D. 0:18 E. 0:24
7. An insurance company sells exactly 3 types of insurance: re, auto, and
ood insurance. Suppose
that every customer who buys
ood insurance also buys re insurance, 20% of the customers buy re
and auto insurance, 40% of customers who buy re insurance also buy
ood insurance, and 25% of
customers who buy auto insurance also buy re insurance. What is the probability that a randomly
chosen customer buys
ood insurance?
A. 0:16 B. 0:25 C. 0:40 D. 0:55 E. 0:80
8. Let X be a random variable with moment generating function
MX(t) =

3 + e2t
4
3
Find the variance of X.
A.
9
16
B.
9
8
C.
9
4
D.
9
2
E. 9
TIA 1/P Seminar p. 2 Sample Exam 4
9. An insurance company categorizes its customers as low risk, medium risk, and high risk. Suppose that
20% are low risk, 50% are medium risk, and 30% are high risk. If 5 customers are chosen at random,
what is the probability that exactly twice as many of them are low risk as high risk given that at least
one customer chosen is high risk?
A. 0:03 B. 0:09 C. 0:11 D. 0:15 E. 0:27
10. The size of a loss due to a hurricane has cdf
F(x) = 1 ??
102
(x + 10)2 ; x > 0:
Suppose that a customer has a deductible of 20. Find the probability that the insurance company's
payment exceeds 10 given that the loss is less than 50.
A. 0:028 B. 0:036 C. 0:064 D. 0:114 E. 0:257
11. The moment generating function of X is given by
MX(t) = :3 + :4e2t + :3e3t:
Find P[X > 1].
A. 0 B. 0:3 C. 0:4 D. 0:6 E. 0:7
12. The number of accidents that occur in a small town in a year satisfy P[N = 0] = p0, P[N = 1] = p1,
and for n 1, P[N = n + 1] = P[N = n]=2. If EN = 1=3, what is P[N = 0]?
A.
1
6
B.
2
6
C.
3
6
D.
4
6
E.
5
6
13. An insurance company with 500,000 customers observes that the probability of a customer ling a claim
in a given year is 2%. If the customers le claims independently of each other, nd the coecient of
variation of the number of claims that are led in a year.
A. 0:01 B. 0:14 C. 7 D. 49 E. 101
14. An insurance company sells 5,000 policies, each with a premium of $110. Each policy will pay $10,000
with probability 1%, and 0 with probability 99%, independently of the other policies. What is the
approximate probability that the total payments on these 5,000 policies will exceed the total premiums?
A. 0:08 B. 0:24 C. 0:49 D. 0:71 E. 0:92
15. Let X and Y be random variables whose joint density is proportional to xy + xy2 for 0 x 1,
0 y 1, and 0 otherwise. Find EY .
A.
7
24
B.
10
24
C.
3
5
D.
7
10
E.
3
4
TIA 1/P Seminar p. 3 Sample Exam 4
16. If X is uniformly distributed on [0; 2], and given that X = x, Y is uniformly distributed on [x; 2x],
what is P[Y 2]?
A. 0:23 B. 0:31 C. 0:50 D. 0:69 E. 0:84
17. Suppose that X and Y are jointly normal random variables, with EX = 1, VarX = 4, and EY = ??2,
E
??
Y 2

= 5. If the correlation of X and Y is ??1=2, what is the probability that the sum of X and Y
is positive?
A. 0:16 B. 0:28 C. 0:37 D. 0:72 E. 0:84
18. Light bulbs for my living room lamp appear to have a random lifetime that is exponentially distributed
with mean 2 weeks. If I replace a light immediately when it burns out, what is the probability that I
will use up at least 3 light bulbs within the next 5 weeks?
A. 0:24 B. 0:46 C. 0:54 D. 0:76 E. 0:99
19. An insurance company feels that the damage that its customers receive has density 3x2=64 for 0
x 4, and 0 otherwise. Currently, the policy has a deductible of 1. If the insurance company adds a
payment limit of 2, how much money can they expect to save per customer?
A. 0:06 B. 0:32 C. 0:48 D. 0:64 E. 0:80
20. The number of hurricanes per month is approximately Poisson distributed, with mean 2 in August, 3
in September, and 1 in October. Assuming that the number of hurricanes in each of the months are
inependent, what is the probability that there will be no more than 3 total hurricanes during this 3
month period?
A. 0:06 B. 0:15 C. 0:27 D. 0:85 E. 0:94
21. If the joint density of X and Y is
f(x; y) =
(
5
16xy2 0 < x < y < 2
0 otherwise,
nd the variance of Y .
A. 0:08 B. 0:28 C. :63 D. 1:67 E. 2:86
22. Suppose that X and Y are independent random variables with common density f(x) = 2e??2x. Find
P[X < 1 j X + Y < 2].
A. 0:79 B. 0:83 C. 0:88 D. 0:91 E. 0:95
TIA 1/P Seminar p. 4 Sample Exam 4
23. Suppose that P[A [ B] + P[A0 [ B] = 2P[B0] and P[AB0] = :2. If A and B are independent, what is
P[A]?
A.
4
15
B.
3
10
C.
2
5
D.
8
15
E.
3
5
24. I have two urns, one of which contains 3 blue balls and 7 red balls, and the other contains 7 blue balls
and 3 red balls. Suppose that I randomly select an urn and then randomly draw two balls from the
urn without replacement. If both are blue, what is the probability that a ball chosen uniformly from
the other urn is also blue?
A. 0:15 B. 0:20 C. 0:35 D. 0:50 E. 0:70
25. Suppose that X and Y are uniformly distributed on the set 0 X 4; 0 Y 4 and X + Y 1,
and let Z = XY . Find P[Z < 4].
A. 0:44 B. 0:48 C. 0:54 D. 0:56 E. 0:58
26. Let X be a random variable with density
f(x) =
(
353
(x+5)4 x > 0
0 otherwise.
What is the median of X?
A. 1:3 B. 2:5 C. 3:6 D. 4:7 E. 5:2
27. Let X and Y be independent continuous uniform random variables, each on the interval (0; 2). If
Z = X + Y and W = X ?? Y , nd E[WZ].
A. ??2 B. 0 C. 1 D. 2 E. 4
28. If the joint density of X and Y is xe??x(y+1) for x; y > 0, what is the expected value of X?
A. 1 B. 2 C. e2 D. 2e2 E. 1 + e2
29. The cdf of X is given by
F(x) =
8>>>><
>>>>:
0 x < 1
x2
8 1 x < 2
x
4 2 x < 3
1 3 x
Find VarX.
A. 0:56 B. 0:62 C. 1:06 D. 2:88 E. 4:96
TIA 1/P Seminar p. 5 Sample Exam 4
30. An actuary estimates that claims from a randomly chosen customer this year have density
f(x) =
2 42
(x + 4)3
and that claims will experience in
ation of 5% next year. What is the probability that the claim from
a randomly chosen customer next year will be greater than 2?
A. 0:44 B. 0:46 C. 0:48 D. 0:50 E. 0:52
TIA 1/P Seminar p. 6 Sample Exam 4
Answers
(1) C
(2) D
(3) B
(4) D
(5) A
(6) E
(7) A
(8) C
(9) C
(10) B
(11) E
(12) E
(13) A
(14) B
(15) D
(16) D
(17) B
(18) B
(19) B
(20) B
(21) A
(22) D
(23) B
(24) C
(25) E
(26) A
(27) B
(28) A
(29) A
(30) B
TIA 1/P Seminar p. 7 Sample Exam 4

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

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