Name: Instructor: François Ledrappier

Math 323-01 Exam II, March 17, 2004

You may use your own calculator and your own textbook.

Part I consists of 6 multiple choice questions. Record your answers by placing an × through one letter for each problem on this answer sheet. Part II consists of 2 partial credit problem. Write your answer and show ALL your work on the page on which the question appears. Do not remove this answer sheet. You will return the whole exam. Each multiple choice question is worth 5 points. Each partial credit question is worth 15 points.

You are taking this exam under the honor code.

1. a  b  c  d  e

2. a  b  c  d  e

3. a  b  c  d  e

4. a  b  c  d  e

5. a  b  c  d  e

6. a  b  c  d  e

Multiple choice points:

Partial credit points:

Total points:

Part I: Multiple Choice Questions (5 points each)

1. Let X be a normal random variable with expectation 3 and variance 4. Rank the following events from most likely to least likely:

A. |X| ≤ 1

B. X ≥ 5

C. X ≤ 0

(a) ABC  (b) ACB  (c) BAC  (d) CAB  (e) CBA

2. Let X be a uniform random variable on some interval (a, b). We know that the expectation of X is 3 and the variance is 3. What is the interval (a, b)?

(a) (0,1)  (b) (2,4)  (c) (0,3)  (d) (0,6)  (e) (.5,5.5)

3. To encourage more people to shop on Mondays, the manager of a small supermarket decides the following promotion: every Monday, he will draw a two digit number at random and all customers with exactly that number of cents on their bill will be refunded 10 % of that bill. If 200 customers visit his store on a particular Monday, what is the probability that he has no winner?

(a) 0.018  (b) 0.135  (c) 0.25  (d) 0.277  (e) 0.367

4. Alice and Bob play rock, scissors, paper. At each game, they choose their bet independently of one another, with equal probability, so that there is a tie with probability 1/3, and each of them has a probability 1/3 of winning. They play 200 independent games. What is the approximate probability that Alice wins less than sixty games?

(a) 0.03  (b) 0.07  (c) 0.14  (d) 0.18  (e) 0.3

5. Which of the following functions can be a density of a continuous random variable?

(A): f(x) = \begin{cases}c(x^{2} − x) & 0 ≤ x ≤ 2 \\ 0 & otherwise \end{cases} (B): f(x) = \begin{cases}sin x + c & −π ≤ x ≤ π \\ 0 & otherwise \end{cases}

(C): f(x) = \begin{cases}sin x + c & −\frac{π}{8} ≤ x ≤ \frac{π}{8} \\ 0 & otherwise \end{cases}

(a) A only  (b) B only  (c) C only  (d) A and B only  (e) A and C only

6. Suppose that, on average, you receive 2 “wrong number” phone calls per week. What is the probability that the next “wrong number” phone call you receive will be within the next 2 days?

(a) .9817  (b) .0183  (c) .2857  (d) .4353  (e) .5647

Part II: Partial Credit Problems (15 points each)

7.(15 points) Charles and David play table-tennis, and they are equally likely to win each point, independently of who serves and independently of the other points already played. The winner of a set is the first one to reach at least 21 points with a two point lead. Let X be the number of points in a set.

(a) Give a formula for the probability that {X = 21 + j}, for j = 0, 1, …, 19.

(b) Assume that after the 40th point, the score is 20/20. What is the distribution of the remaining number of points, namely P (X = 40 + 2j| score is 20/20), for j = 1, 2 . . ..

(c) What is the probability that {X ≥ 50}?

8. (15 points) Let X be a continuous random variable representing the lifetime of a prehistoric man. Assume that the hazard rate function λ(t) for X is given by

λ(t) = \begin{cases}\frac{t}{200} & 0 ≤ t \\ 0 & t < 0 \end{cases} .

(a) Find P {X > 20}.

(b) Find the expected lifetime of a prehistoric man.