Closed book. Two pages of hand-written notes, front and back. No calculator. 6 minutes. Cover page and four pages of exam. Four questions. To receive full credit, show enough work to indicate your logic. Do not spend time calculating. Unless stated otherwise, you will receive full credit if someone with no understanding of probability could simplify your answer to obtain the correct numerical solution. This exam covers through Chapter 5 of Solberg (August 25). Recall: The Poisson pmf with meanµis f (x )=e µ µ x / x! for x =, 1, 2,... Recall: The geometric pmf with probability of success p is f (x )=p (1 p) x 1 for x = 1, 2, 3,... The mean is 1 / p. Recall: The exponential pdf with mean 1 /λ is f (x )=λe λx for x. Recall: The exponential cdf with mean 1 /λ is F (x )=1 e λx for x. Recall: a c 1 b d = d /(ad bc ) c /(ad bc ) b /(ad bc ) a /(ad bc ) Score Exam #3, Spring 26 Schmeiser
Closed book. Two pages of hand-written notes, front and back. No calculator. 1. (Based on Solberg s Problem 9, Chapter 5) Customers arrive at a bank machine according to a Poisson process with rate 4 per hour. Over time, half of the customers are men and half are women. Suppose that twenty men arrive in a particular eight-hour day. (a) (six points) Conditional that the twenty men arrived, determine the arrival rate for women for that particular eight-hour day. λ W =λ/2=2women per hour (b) (six points) Conditional that the twenty men arrived, determine the expected number of women who arrive in that particular eight-hour day. λ W t = (2) (8)=16 women (c) (seven points) Conditional that the twenty men arrived, determine the probability that twenty women arrive in that particular eight-hour day. Let W denote the number of women arriving in a random eight-hour day. Arrivals are according to a Poisson process, so the number of arrivals is Poisson: P(W = 2)= e µ µ 2, 2 whereµ=λ W t = 16 from Part (b). HP (d) (seven points) Conditional that the twenty men arrived, determine the distribution of time from the beginning of the day until the first woman arrives. The distribution is exponential with mean 1 /λ W = 1 / 2 hour. Exam #3, Spring 26 Page 1 of 4 Schmeiser
2. (Based on Solberg s Problem 2, Chapter 5) Two copies of a particular library book are held on reserve. Anyone who wants to read the book has to request it at the reserve desk and must return it there the same day; the copy cannot be removed from the library. Demands for the book occur throughout the day according to a Poisson process with rate two per hour. Users keep the book for an exponential time, with an average of ninety minutes; user times are independent of each other. At opening time, both copies are available for check out. (a) (six points) Which seems to be the more-appropriate assumption: that demands are from a Poisson process or that time to keep the book is exponential? Why? To argue against Poisson arrivals with constant rate: dependence: students arrival together dependence: finite number of students reduces demand rate nonstationary: demand rate should depend on time of day To argue against exponential time to keep the book: mode: zero is not the most likely time memory: remaining time depends on past time finite range: time must be less than time until closing (b) (eight points) State a Markov process model for this situation. Let time be measured in hours. (Other choices, such as minutes, are ok.) Let state be the number of copies available. (Number checked out is also ok.) Then the rate matrix is 2 2 8 / 3 4 / 3 2. 4 / 3 Λ = 1 2 / 3 2 (c) (three points) Is your model of Part (b) a birth-death model? < yes > (d) (six points) Write the Kolmogorov forward equations, including initial conditions. Using double subscripts: The initial state is, so p ()=1, p 1 ()=and p 2 ()=. For j =, 1, 2, dp j (t ) = p + p 1 (t )λ 1j + p 2 (t )λ 2j dt Alternatively, use a single subscript: The initial state is, so p ()=1, p 1 ()=and p 2 ()=. For j =, 1, 2, dp j (t ) = p + p 1 + p 2 dt (e) (six points) In terms of your notation from Part (d), what is the probability that no copies are available four hours into the day? (with two subscripts) p 2 (4) or (with one subscript) p 2 (4) Exam #3, Spring 26 Page 2 of 4 Schmeiser
3. Consider the three-state rate matrix 1 2 3.5 1..1 3..4 2.. Assume that at time t = 6 the system is equally likely to be in any of the three states. (a) (eight points) From time t = 6, determine the expected time until the system is next in State 1. (If in State 1 at time t = 6, wait until the next visit.) Maybe never return to State 1, so this is not a mean first-passage time. Either of two answers for full credit: Expected time is infinite. Conditional mean first-passage time: h 1 + h 2 = ( 1 /λ 11 )+( 1 /λ 22 )=2+(1 / 3)=7 / 3 time units. (b) (six points) From time t = 6, determine the probability that the system ever reaches State 3. State 3 is absorbing, so the answer is one. Because State 3 is absorbing, ok to solve for the steady-state probabilityπ 3 = 1. (c) (six points) Now assume that the system is in State 1. What is the probability that it will next be in State 3? λ 13.4 = λ 12 +λ 13.1+.4 =.8 Exam #3, Spring 26 Page 3 of 4 Schmeiser
4. Consider the time until the next event in a Poisson process with rateλ. (Recall: The time is exponential with mean 1 / λ.) (a) (eight points) Develop a Markov process model that can be used for Parts (b) and (c). Full credit for any of several models. Model 1: Two states, say and 1, with rateλ 1 =λ. Model 2: Infinite states, say, 1,..., with ratesλ i,i+1 =λ. Model 3: Choose the model of your choice. (b) (six points) Based on your model, explain how to calculate the probability that the time until the next event is greater than 1 /λ. Assume currently in State i = for Model 1 or any State i for Model 2. Time in State i is exponential with mean 1 /λ, so P(T > t )=e λ t. Therefore, P(T > 1 /λ)=e 1. More completely, dp i,i (t ) / dt = p i,i (t )λ with p i,i ()=1 implies that p i,i (t )=e λ t. But P(T > t )=p i,i (t ), so the holding time is exponential with rateλ. (c) (six points) Based on your model, explain how to calculate the expected time until the next event. Assume currently in State i = for Model 1 or any State i for Model 2. For any Markov process, the mean holding time is h i = 1 /λ ii = 1 / ( λ)=1/λ. Exam #3, Spring 26 Page 4 of 4 Schmeiser