Exam 7 Solutions

7/27/2019 Exam 7 Solutions

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1/P Sample Exam 7

1. Let X be a Poisson random variable with second moment 6. Find P[X = 3].

A. 0.06 B. 0.12 C. 0.18 D. 0.20 E. 0.22


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2. Let X and Y have joint probability density function

f(x, y) =

3

32(x2 + y2) 0 < x < 2, 0 < y < 2

0 otherwise.

Find the expected value of the maximum of X and Y.

A. 0.6 B. 0.8 C. 1.0 D. 1.2 E. 1.6


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3. Let X and Y have joint probability density function

f(x, y) =

1

5x 0 < y < x < 5

0 otherwise.

Find E[X | Y = 3].

A. 3.7 B. 3.9 C. 4.0 D. 4.1 E. 4.3


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4. Suppose that Z be a normal random variable with mean 5 and variance 2. Let fZ(z, ) denote the

density ofz for a given , and let FZ(z, ) the CDF ofZ for a given . Which of followings is increasing

as increases?

A. fZ(5, )

B. FZ(3, )

C. FZ(8, )

D. The 25th percentile of Z

E. None of the above


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5. An insurance company sells exactly two types of insurance, home owners and auto insurance. 55% of

their customers have homeowners insurance and 30% have both types of insurance. What fraction of

the customers have auto insurance?

A. 15% B. 25% C. 70% D. 75% E. 85%


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6. A small shipping company has 5 trucks. The number of accidents that each truck has per year has

a Poisson distribution with mean 1.5. If each accident costs the company $250, and the number of

accidents per truck are independent, what is the standard deviation of the annual cost of accidents to

the shipping company?

A. 685 B. 839 C. 1022 D. 1286 E. 1531


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7. An electronics dealer stocks laptops in 3 colors: black, red, and blue. They currently have 14 laptops in

stock, 8 of which are black, 4 of which are red, and two of which are blue. If three laptops are chosen

at random, what is the probability that at least one of them is red?

A. 0.57 B. 0.60 C. 0.64 D. 0.67 E. 0.70


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8. An unfair coin that has probability p of being heads is flipped. If the coin comes up heads, a fair die

whose faces are 0, 1, 2, 3, 4, and 5 is rolled, while if it is tails, a standard fair die is rolled. Let N be the

number displayed on the die. If FN(3) = 17/30, then what is FN(4)?.

A.

18

30 B.

19

30 C.

20

30 D.

21

30 E.

22

30


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9. Suppose that X, Y and Z are independent exponential random variables with means , and 4 respec-

tively. Let U = X+ 2Y + 3Z and V = X Y. Suppose that EU = 24 and VarV = 45.

What is ?

A. 2 B. 3 C. 4 D. 5 E. 6


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10. Let T denote the lifetime of a citizen of Freedonia, and suppose that the density of T is

f(t) =

2t

10020 < t < 100

0 otherwise.

What is the probability that a 20 year old Freedonian will live to be 40?

A. 0.80 B. 0.84 C. 0.88 D. 0.92 E. 0.96


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11. Suppose that the moment generating function ofX is (12t)5, and that Y is an independent random

variable with a Gamma distribution with mean 6 and variance 18. Which of the following is the moment

generating function for X+ Y?

A. 1

1 2t5 1

1 3t2

B.

1

1 2t

5+

1

1 3t

2

C.

1

1 2t

5

1

1 2t

3

D.

1

1 2t

5

1

1 2t

3

E.

1

1 2t

5

+

1

1 2t

3

TIA 1/P Seminar p. 11 Sample Exam 7


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12. The amount of time T that lightbulbs last has a density of

f(t) =3 503

(x + 50)4t > 0.

Suppose that I turn on 5 different lightbulbs. What is the expected amount of time until the first one

fails?

A. 3.3 B. 3.6 C. 4.2 D. 4.5 E. 5.0


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13. At a dinner party, 6 men and 2 women sit at a circular table. If seats are randomly assigned, what is

the probability that the two women are not seated next to each other?

A. 0.57 B. 0.62 C. 0.66 D. 0.71 E. 0.75


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14. The joint density of X and Y is proportional to x2y for 1 x y 2 and 0 otherwise. Find the

expected value ofXY.

A. 2.14 B. 2.52 C. 2.87 D. 3.35 E. 3.95


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15. Losses under an insurance policy are uniformly distributed on the interval [0, 1000]. The policy has a

deductible of 400, and the insurance company only reimburses policyholders by p times the amount by

which the loss exceeds the deductible. If the average payment per loss is 108, what is p?

A. 0.27 B. 0.36 C. 0.48 D. 0.60 E. 0.72


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16. The failure time for a machine has density f(t) = 4te2t for t > 0. What is the probability that the

machine will fail between time 1 and time 3?

A. 0.13 B. 0.21 C. 0.28 D. 0.35 E. 0.39


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17. Suppose that X an exponential random variable such that P[X 3] = p. What is P[X > 12|X > 3]?

A. 0.5 B. p C. p2 D. p3 E. p4


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18. Let X, Y and Z be iid Poisson random variables with mean , and let U = 75X+ 100Y+ 25Z. If the

coefficient of variation ofU is 0.7, what is ?

A. 0.71 B. 0.83 C. 0.95 D. 1.05 E. 1.18


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19. An urn has 13 balls: 9 white, 2 red and 2 blue. If I randomly choose 5 of the balls without replacement,

what is the probability of getting at least one ball of every color?

A. 0.26 B. 0.29 C. 0.32 D. 0.35 E. 0.38

TIA 1/P Seminar Sample Exam 7


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20. The number of losses N has a binomial distribution with mean 5/3 and variance 10/9. If each loss size

is 2, and there is a deductible of 3, what is the expected payment?

A. 1.1 B. 1.6 C. 2.3 D. 2.7 E. 3.3


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21. Let X and Y have joint probability density function f(x, y) = 19

for 0 < x < y2 < 9 and y > 0, and 0

otherwise. Find the variance of X given that Y = 2.

A.1

6B.

3

4C.

4

3D.

25

12E.

27

4


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22. Suppose that M(t) is a function such that M(0) = 1, M(0) = 1, and M(0) = 4. If Y is a random

variable with moment generating function [M(t)]2, then what is the variance of Y?

A. 6 B. 8 C. 9 D. 10 E. 15


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23. An insurance company has two types of risks, A and B. The probability of a loss and the size of the

resulting loss for each type of risk is given below:Risk P[Loss] Size of Loss

A 0.2 300

B 0.1 100If 40% of the risks are of type A, what is the variance of the actual losses that occur?

A. 6, 900 B. 7, 800 C. 8, 200 D. 8, 800 E. 9, 800


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24. The density of X is given by

f(x) =

110e

(x)/10 x >

0 otherwise

for an unknown > 0. If EX = 12, what is ?

A. 1 B. 2 C. 4 D. 5 E. 10


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25. An insurance company offers 3 types of insurance: Fire, earthquake, and hurricanes. You are given the

following:

No customer has both hurricane and earthquake insurance.

25% of those with earthquake coverage also have fire insurance. 50% of those with 2 types of insurance have hurricane insurance.

50% of the customers have fire insurance.

The number of customers who have hurricane insurance only is the same as the number of customers

with 2 types of insurance.

What fraction of the customers have only 1 type of insurance?

A. 0.2 B. 0.4 C. 0.5 D. 0.6 E. 0.8


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26. The density ofX is proportional to 1 x2 for 1 x 1 and is 0 otherwise. If Y = X2, then what

is EY?

A. 0.1 B. 0.2 C. 0.3 D. 0.4 E. 0.5


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27. Suppose that M(t) is the moment generating function for a random variable X. Which of the following

are also moment generating functions for some random variable?

(i) 3M(t)

(ii) [M(t)]

3

(iii) M(3t)

A. (i) only

B. (ii) only

C. (iii) only

D. (i) and (ii) only

E. (ii) and (iii) only


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28. The joint density of X and Y is given by

f(x, y) =

40x5y3 0 < y < x < 1

0 otherwise.

What is P[XY > 1/2]?

A. 0.69 B. 0.73 C. 0.77 D. 0.81 E. 0.85


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29. Suppose that N is a Poisson random variable with mean 1.5. The mean absolute deviation of a random

variable X is E|X |, where denotes the mean ofX. What is the mean absolute deviation of N?

A. 0.0 B. 0.5 C. 1.0 D. 1.5 E. 2.0


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30. The cdf ofX is given by

F(x) =

0 x < 1x4

1 x < 2

1 1x2

2 x <

What is EX?

A. 1.1 B. 1.6 C. 2.1 D. 2.5 E. 3.2

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Exam 7 Solutions