Mathematics 571 Introduction to Probability Models Spring 2007
Page under construction, February 2007

Each problem is weighted as 1 or 2 points. Undergraduates and auditors must turn in 10 points worth of problems in each assignment. Graduate students must turn in 15 points worth of problems in each assignment. Submissions beyond the requirement will be evaluated but not count toward your grade. There are no extra credit assignments.
Homework Style Guide Since there are quite a few students in this course, I would appreciate it if you followed these guidelines.
  1. Due Wednesday, January 24: Read Chapter 4, Sections 1 and 2.
  2. Due Monday, January 29: Read Chapter 4, Section 3.
  3. Due Wednesday, January 31:
    1. Read Chapter 4, Section 3
    2. One point problems: 1, 2, 3, 5, 7, 8, 9, 10, 11
    3. Two point problems: 4, 6, 12, 13
  4. Due Monday, February 5: Read Chapter 4, Section 3
  5. Due Wednesday, February 7:
    1. Read Chapter 4, Section 4
    2. Any one point problem you did not submit on the last assignment
    3. One point problems: 14, 21a, 22
    4. One point: Determine the transition matrix for the following Markov chain: Consider the number of tosses of a coin with probability p of heads. After n independent tosses of the coin the state is the number of heads minus the number of tails.
    5. One point: Determine the transition matrix for the following Markov chain: A rat is placed in a square maze consisting of nine identical square compartments arranged in three rows of three squares. Each compartment is connected directly to any compartment with which it shares a side, so that the rat may move left and right and up and down, but not diagonally. The rat moves from compartment to compartment at random, and the state of the chain is the compartment the rat is in.
    6. One point: Consider two urns A and B containing a total of N balls. An experiment is performed in which a ball is selected at random at time t (t = 1, 2, ...) from among the N balls (all balls have the same chance of being selected). Then an urn is selected at random where urn A is selected with probability p and urn B with probability q = 1-p, and the ball that was selected is placed into the randomly chosen urn. The state of the system is the number of balls in urn A. What is the transition matrix of this Markov Chain and what are the equivalence classes.
    7. One point: Continuation. Now assume that at time t there are k balls in urn A. At time t+1 an urn is selected in proportion to the number of balls it contains, so that urn A is selected with probability k/N and urn B with probability (N-k)/N. Then a ball is selected from urn A with probability p and from urn B with probability q = 1-p and placed in the previously selected urn. Again, the state of the system is the number of balls in urn A. What is the transition matrix of this chain and what are the equivalence classes?
    8. One point: Continuation. Now assume that at time t an urn and a ball are chosen with the probability proportional to the number of balls in the urn (i.e. a ball is chosen from urn A with probability k/N and from urn B with probability (N-k)/N, and urn A is chosen with probability k/N and urn B with probability (N-k)/N if at time t A contains k balls), and the choice of the ball is independent of the choice of the urn. If the state of the chain is the number of balls in urn A, what is transition matrix and what are the equivalence classes?
    9. Any two point problem you did not submit on the last assignment
    10. Two point problems: 15, 16, 17
    11. Suppose that A is a two by two matrix whose rows sum to 0. Show that there is a constant a so that the nth power of A is the (n-1)th power of a times A itself.
    12. Two points: Suppose that a Markov chain has two states, 0 and 1. Suppose that
      • P(X(n+1) = 1 | X(n) = 0) = p
      • P(X(n+1) = 0 | X(n) = 1) = q.
      Find the P(X(n) = 1 | X(0) = 1).
  6. Due Monday, February 12: Read Chapter 4, Section 5
  7. Due Wednesday, February 14:
    1. Read Chapter 4, Section 06.
    2. One point problems: Any of the following that have not been submitted previously: 19, 20, 21, 22, 23, 24, 27, 29, 30, 31, 32, 33, 34, 35
    3. Two point problems: 25, 26, 28, 36, 37, 38, 39, 40, 42, 43, 44, 45, 46.
  8. Due Monday, February 19: Read Chapter 4, Section 6
  9. Due Wednesday, February 14:
    1. Read Chapter 4, Section 07.
    2. One point problems: The exercises on the webpage Moment Generating Functions .
    3. One point problems from your text: Any previously assigned one point problems that you have not done.
    4. New one point problems: 54, 55, 56, 58, 59, 60
    5. Two point problems: 61
    6. Additional two point problems
  10. Due Monday, February 26: Read Chapter 5, Section 1 and 2
  11. Due Wednesday, February 28:
    1. Read Chapter 5, Section 2.
    2. One point problems from your text: Any previously assigned one point problems that you have not done.
    3. New one point problems: 63, 64, 66, 67a-d
    4. Additional one point problems
    5. Two point problems: 65, 67e-g,
    6. Additional two point problems
  12. Due Monday, March 5: Read Chapter 4, Sections 1 through 7. Come prepared with questions on the chapter.
  13. Due Wednesday, March 7:
    1. Exam on Chapter 4, Sections 1 through 7.
    2. Homework: Review assignment for the exam
  14. Due Monday, March 12: Read Chapter 5, Sections 3.1 through 3.3.
  15. Due Wednesday, March 14: Reach Chapter 5, Sections 3.4 through 3.6
  16. Due Monday, March 26: Read Chapter 5, Section 4.
  17. Due Wednesday, March 28: Reach Chapter 6.1 to 6.3.
    1. Homework: One point problems: 2, 4, 9, 18, 31, 36, 37, 38, 39, 59
    2. Homework: Two point problems: 40, 42, 45, 46, 53, 66
  18. Due Monday, April 2: Read Chapter 6, Section 4 and 5. In 6.4 you can ignore all the differential equation material and read the handout on the webpage instead.
  19. Due Wednesday, April 4: Read Chapter 6, Section 8.
    1. Homework: Any one point problem from the last assignment that was not already submitted.
    2. Homework: Any two point problem from the last assignment that was not already submitted.
    3. Homework: One point problems: 5.77, 5.78,
    4. Homework: Two point problems: 5.68, 5.69, 5.71, 5.73, 5.82
    5. Homework: One point problems: 6.1, 6.2, 6.3, 6.4, 6.12, 6.13, 6.14, 6.15
  20. Due Monday, April 9: Read Chapter 6, Section 4 and 5. In 6.4 you can ignore all the differential equation material and read the handout on the webpage instead.
  21. Due Wednesday, April 11:
    1. Homework
  22. Due Monday, April 16: Chapter 10, Sections 1-3.
  23. Due Wednesday, April 18:
    1. Read Chapter 10, Sections 1-3
    2. Homework
    3. Homework 9 with solutions of most problems.
  24. Due Monday, May 7:
    1. Homework

You may be required to re-submit homework problems until you have solved them correctly if you are to get credit for them. Each problem will be assigned a letter grade.