Name: Instructor: François Ledrappier

Math 323-01 Final Exam, May 3, 2004

You may use your own calculator, your own textbook, and your notes. You may use nothing else. This test consists of 15 multiple choice questions (5 points each). Record your answers by placing an × through one letter for each problem on this answer sheet. Do not remove this answer sheet. You will return the whole exam. You are taking this exam under the honor code.

1. a  b  c  d  e  9. a  b  c  d  e
2. a  b  c  d  e  10. a  b  c  d  e
3. a  b  c  d  e  11. a  b  c  d  e
4. a  b  c  d  e  12. a  b  c  d  e
5. a  b  c  d  e  13. a  b  c  d  e
6. a  b  c  d  e  14. a  b  c  d  e
7. a  b  c  d  e  15. a  b  c  d  e
8. a  b  c  d  e

Total points:

1. Let X and Y be two Bernoulli random variables ( i.e. X and Y take only the values 0 and 1). Assume that:

P (X = 1) = \frac{1}{3}, P (X + Y = 1) = \frac{1}{2}, and X and Y are independent.

What is the probability that Y = 0?

(a) \frac{1}{6}  (b) \frac{1}{3}  (c) \frac{1}{2}  (d) \frac{2}{3}  (e) \frac{3}{4}

2. A box contains one chocolate ball and ten plastic balls. You roll a fair die, and the number obtained tells you how many balls you can pick from the box. What is the probability that you get the chocolate ball?

(a) \frac{6}{11}  (b)\frac{7}{22}  (c)\frac{7}{11}  (d) \frac{1}{2}  (e) \frac{14}{33}

3. A shooter hits the target with probability 65%. Suppose that he needs 3 hits to win a prize. Find the expected number of attempts he needs to make in order to win the prize.

(a) 4.6  (b) 1.9  (c) 4.9  (d) 7.2  (e) 3.0

4. Suppose that in a week there are, on average, 7 accidents at an intersection. What is the probability that at least one accident will happen today?

(a) 63%  (b) 14%  (c) 50%  (d) 54%  (e) 27%

5. A shooter aims at a target in a practice room. He can obtain 0, 1, 2, or 3 points at each shot. The results are independent from each other, and each time he scores: 3 points with probability 0.5, 2 points with probability 0.3, 1 points with probability 0.1, 0 points with probability 0.1. He runs a series of 100 shots. What is the probability that his total score is at least 240? (You will have to use a continuity correction in this problem.)

(a) .0122  (b) .0183  (c) .0233  (d) .0405  (e) .09

6. In a group of 30 people, 20 read newspapers and 13 read magazines. Suppose that every person in this group reads something (either a magazine, or a newspaper, or both). What is the probability that a person, randomly chosen from this group, reads newspapers but does not read magazines?

(a) .67  (b) .85  (c) .65  (d) .15  (e) .57

7. Inside the package of a brand of corn-flakes, one finds one of 10 different baseball cards. How many packages should one buy so that the expectation of the number of distinct cards is at least 7?

(a) 9  (b) 10  (c) 11  (d) 12  (e) 13

8. A high school reunion was attended by a total of 45% of women and 32% of men from a former graduating class. If 60% of of the original class were male, what was the percentage of women at the reunion?

(a) 40%  (b) 18%  (c) 45%  (d) 48%  (e) 50%

9. The radius R of a circle is a random variable with the expectation E[R] = 3 and variance V ar(R) = 4. What is the expected value of the area of the circle?

(a) 3π  (b) 13π  (c) 7π  (d) 4π  (e) cannot be determined

10. The hazard rate function for a light bulb is

λ(t) = \begin{cases}(\frac{t}{100})2 & 0 ≤ t ≤ 25 \\ \frac{t}{1,000} & 25 < t \end{cases} .

What is the probability that the bulb is working after 50 hours?

(a) .05  (b) .23  (c) .90  (d) .10  (e) .77

11. What is the value of a such that the following function f is a density?

f(x) = \begin{cases}a − x^{2} & −1 ≤ x ≤ 1 \\ 0 & otherwise \end{cases}

(a) \frac{1}{2}  (b)\frac{1}{4}  (c)\frac{1}{6}  (d) 1  (e) none.

12. Classes A and B consist of 10 boys and 10 girls each. The students from both classes are randomly paired into boy–girl couples for a dance contest. Let N be the number of couples where both students are from the class A. What is the variance of N?

(a) 1.00  (b) 1.23  (c) 1.32  (d) 1.50  (e) 2.00

13. Let X be a random variable with moment generating function M(t) given by:

M(t) = exp ((\frac{λ}{λ − 1})^{t} − 1) .

What is the variance of X?

(a) log \frac{λ}{λ − 1}  (b)(log \frac{λ}{λ − 1})^{2}  (c)(\frac{λ}{λ − 1})^{2}  (d) \frac{λ}{λ − 1}  (e) λ^{2}

14. Let X_{1}, …, X_{64} be independent exponential random variables with mean \frac{1}{2}. Use the Central Limit Theorem to approximate P {X_{1} + ⋯ + X_{64} > 36}.

(a) .0158  (b) .8413  (c) .2541  (d) .9842  (e) .1587

15. Let X and Y be jointly distributed continuous random variables with the density

f(x, y) = \begin{cases}2e^{−x}e^{−2y} & 0 < x < ∞, 0 < y < ∞ \\ 0 & otherwise \end{cases} .

Which of the following integrals represents the probability P \{X − \frac{1}{2}Y > 2\}?

(a) ∫_{2}^{∞} ∫_{0}^{∞} 2e^{−x}e^{−y}dydx

(b) ∫_{0}^{∞} ∫_{0}^{∞} 2e^{−x}e^{−2y}dydx

(c) ∫_{2}^{∞} ∫_{0}^{2x−4} 2e^{−x}e^{−2y}dydx

(d) ∫_{2}^{∞} ∫_{0}^{2x−4} 2e^{−x}e^{−y}dydx

(e) ∫_{0}^{∞} ∫_{0}^{2x−4} 2e^{−x}e^{−2y}dydx