Name: Instructor: François Ledrappier

Math 323-01 Exam III, April 14, 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_{i}, i = 1, 2, 3 be three independent Normal Random Variables with respective means μ_{1} = 1, μ_{2} = 0 and μ_{3} = −1 and variances σ_{1}^{2} = 4, σ_{2}^{2} = 1 and σ_{3}^{2} = 2. What is the value of

P (2X_{1} − X_{2} + 4X_{3} > 5)?

(a) .0668  (b) .1469  (c) .1587  (d) .2568  (e) .3085

2. Let X, Y be Bernoulli Random Variables (i.e. X and Y take only the values 0 and 1 ) such that E(X) = 1/2, E(Y) = 2/3. What is the range of possible values of Cov(X, Y)?

(a) [0, \frac{1}{3}]  (b) [−\frac{1}{2}, \frac{1}{2}]  (c) [−\frac{\sqrt{2}}{3}, \frac{\sqrt{2}}{3}]  (d) [−\frac{1}{6}, \frac{1}{6}]  (e) [−\frac{\sqrt{2}}{6}, \frac{\sqrt{2}}{6}]

3. Let X and Y be two independent continuous Random Variables with respective densities f_{X}, f_{Y} given by:

f_{X}(x) = \begin{cases}\frac{x}{2} & 0 ≤ x ≤ 2 \\ 0 & otherwise \end{cases} f_{Y} (x) = \begin{cases}1 & 0 ≤ x ≤ 1 \\ 0 & otherwise. \end{cases}

Let f_{Z} be the density of the variable Z = X + Y. What is the value of f_{Z}(2)?

(a) \frac{1}{2}  (b)\frac{3}{4}  (c)\frac{2}{3}  (d) 1  (e) \frac{5}{4}

4. Let X, Y, Z be three Random Variables such that the point with coordinates (X, Y, Z) in the usual euclidean space is uniformly distributed on the cylinder C:

C = {(x, y, z) : x^{2} + y^{2} ≤ 1 and 0 ≤ z ≤ 1}.

Which of the following statement is true?

(a) X is independent of Y, and X is independent of Z

(b) Y is independent of Z, and Y is independent of X

(c) X is independent of Z, and Y is independent of Z

(d) all of the above

(e) none of the above.

5. An urn contains 20 balls, some of which are black and some are white. The number of white balls is a binomial random variable with parameters (20, p), where p is a uniform random variable on [0, 1]. Suppose that 4 balls are randomly chosen from the urn and let N be the number of white balls chosen. What is E[N]?

(a) 0  (b) 1  (c) 2  (d) 3  (e) 4.

6. On a farm, 30 rabbits, of which 15 are male and 15 are female, are randomly put in 15 cages, two to each cage. Let N be the number of male-female couples. What is E[N]?

(a) \frac{15}{2}  (b) \frac{15}{29}  (c) \frac{300}{29}  (d) \frac{450}{29}  (e) \frac{225}{29}

Part II: Partial Credit Problems (15 points each)

7. We consider a one-dimensional golf player. When she starts from a point A on the real line and aims at another point B, we assume that the point reached by the ball is on the same line and is given by a Random Variable X with mean B and with variance σ^{2}|B − A|^{2}.

Our golfer starts from the origin and gets at the point 500 in two hits: first she aims at the point 300. She arrives at a point X_{1}. From X_{1}, she aims at the point 500 and arrives at the point X_{2}

a. What the conditional mean and the conditional variance of X_{2}, given that X_{1} = x?

b. What are the mean and the variance of X_{2}?

8. In the situation of the problem 6, assume that in one month each male-female couple brings M bunnies, where M is a random variable with expectation E[M] = 6 and variance V ar(M) = 2. Let X be the number of rabbits on the farm after one month.

(a) Find E[X].

(b) Let N be the random variable from Problem 6. Find V ar(N).

(c) Using the answer to part (b) of this problem, find V ar(X). What assumption should you make?