IE 336 Seat # Name (clearly) Closed book. One page of hand-written notes, front and back. No calculator. 60 minutes.

Transcription

1 Closed book. One page of hand-written notes, front and back. No calculator. 6 minutes. Cover page and four pages of exam. Fifteen questions. Each question is worth seven points. To receive full credit, show enough work to indicate your logic. Do not spend time calculating. You will receive full credit if someone with no understanding of probability could simplify your answer to obtain the correct numerical solution. This test covers through Chapter 4 of Solberg (August 25). Recall: The Poisson pmf with meanµis f (x )=e µ µ x / x! for x =,, 2,... Recall: The geometric pmf with probability of success p is f (x )=p ( p) x for x =, 2, 3,... Score Exam #2, Fall 25 Schmeiser

2 Closed book. One page of hand-written notes, front and back. No calculator. For Questions 4, consider the example of trucks being assigned sequentially to Atlanta, Boston, Chicago, and Detroit, with time incrementing with each the trip between cities. Assume that the one-step transition matrix is P = Atlanta Boston Chicago Detroit a b c.. Determine the values of a, b, and c. 2. Sketch the corresponding digraph, showing all states and one-step transition probabilities. 3. Which states are transient? Which states are absorbing? 4. If the truck is now equally likely to be in any of the four cities, what is the probability that it will next be in Boston? Exam #2, Fall 25 Page of 4 Schmeiser

3 For Questions 5 8, consider the two-state example of the employee being absent or present at work, with time measured by work days. The one-step transition matrix and infinite-step transition matrices are Absent.2 P = Present P ( ) Absent /7 = Present /7 6/7 6/7. We know that the employee is ill; she is absent today and was absent yesterday. 5. In the long run, what fraction of days will the employee be absent? 6. Explain a method to determine the elements of P ( ). 7. Let X denote the number of future days that the employee will be absent during this illness. What do we know about X? 8. Discuss the use of the Markov assumption in this model. (In the context of this model, what does the assumption say? What arguments can you provide to support the assumption? What arguments can you provide against using the assumption? Use the back of this page if you wish, but a long answer is not necessary.) Exam #2, Fall 25 Page 2 of 4 Schmeiser

4 For Questions 9 2, consider Solberg s production-process example, where each transition has the part moving to the next process (Operation, Operation 2, or Inspect) or to a final state (Finished Part or Rejected Part). Parts always begin at Operation. The one-step transition matrix and the hit matrix are P= Operation Operation 2 Inspect FinishedPart RejectedPart E=(I Q) = Operation Operation 2 Inspect E is defined in terms of Q. Write Q.. What is the expected number of times that a part is inspected?. Given that the part is currently being inspected, what is the expected number of additional inspections that will be needed? 2 What fraction of parts at Operation 2 are eventually rejected? Exam #2, Fall 25 Page 3 of 4 Schmeiser

5 For Questions 3 4, consider the equation p ij (n ) = n Σ k= f ij (k ) pjj (n k ), where p ij (n ) is the n -step transition probability of going from state i to state j and f ij (n ) is the probability of going from state i to state j for the first time in n transitions. 3. What is the value of p ij () when i = j? When i =/ j? (2) () () 4. When n = 2, the equation becomes p ij = f ij pjj + f (2) () ij pjj. Explain why the equation is true. (You can use words or math.) 5. Consider a game that ends when one team is ahead by two points. Team A wins the next point with probability.7 if it won the previous point; otherwise it wins the next point with probability.6. Team B has just won a point to even the game. State a Markov chain that could be used to find the probability that Team A will win. Exam #2, Fall 25 Page 4 of 4 Schmeiser