Statistics 253/317 Introduction to Probability Models Winter 2014 - Midterm Exam Friday, Feb 8, 2013 Student Name (print): (a) Do not sit directly next to another student. (b) This is a closed-book, closed-note examination. You should have your hand calculator and one page of formula sheet that you may refer to. (c) You need to show your work to receive full credit. In particular, if you are basing your calculations on a formula or an expression (e.g., E(Y X = k)), write down that formula before you substitute numbers into it. (d) If a later part of a question depends on an earlier part, the later part will be graded conditionally on how you answered the earlier part, so that a mistake on the earlier part will not cost you all points on the later part. If you can t work out the actual answer to an earlier part, put down your best guess and proceed. (e) Do not pull the pages apart. If a page falls off, sign the page. If you do not have enough room for your work in the place provided, ask for extra papers, label and sign the pages. Question Points Available Points Earned 1 10 2 20 3 35 4 35 TOTAL 100 1
Problem 1. [10 points] A Markov chain with state space {0, 1, 2, 3, 4} has transition matrix P = 0 1 2 3 4 0 0 1/2 0 0 1/2 1 0 0 1 0 0 2 0 1/2 0 1/2 0 3 0 0 1 0 0 4 1/2 0 0 1/2 0 Find all communicating classes. For each state, determine its periods, and whether it is recurrent or transient. Explain your reasoning. 2
Problem 2. [20 points] Let i and j be 2 states of a Markov chain. Let f ij be the probability that starting in i the chain ever reaches j and let f ji be the probability that starting in j the chain ever reaches i. (a) [10 pts] If i and j are both transient and i communicates with j, argue that f ij and f ji cannot both be equal to one. (b) [10 pts] If i and j are both recurrent, is it possible for i to be accessible from j but j not accessible from i? Justify your answer. 3
Problem 3. [35 points] [Nearly Symmetric Random Walk] Let {X n : n 0} be a (non-simple) random walk on {0, 1, 2,...} with transition probabilities P i,i+1 = a i+1 a i + a i+1, for i 0 and P i,i 1 = a i a i + a i+1 for i 1, P 0,0 = a 0 a 0 + a 1. Here {a i > 0 : i = 0, 1, 2,...} is a positive sequence such that lim i a i+1 /a i = 1. Observed that P i,i+1 1/2 when i is large. Thus the process behaves like a symmetric random walk when it is far from 0. Since a i > 0 for all i = 0, 1, 2,... and P 0,0 > 0, {X n } is irreducible and aperiodic. We know that a symmetric random walk is null recurrent, will a nearly symmetric random walk be null recurrent or positive recurrent? (a) [10pts] Show that {X n } has a limiting distribution (and hence is positive recurrent) if i=0 a i <. Express the limiting distribution of {X n } in terms of {a 0, a 1, a 2,...}. (Hint: Solve the detailed balanced equation for the limiting distribution.) 4
(b) [5 pts] For a i = 1/(i + 1) 2, i = 0, 1, 2,..., find the limiting distribution of {X n }. You may use the identity n=1 (1/n2 ) = π 2 /6. 5
(c) [10 pts] Let M i,j be the mean time to reach state j starting in state i. First show that for a general random walk on {0,1,2,... }, M i,i+1 = 1 + P i,i 1 M i 1,i + P i,i 1 M i,i+1, for i 1 and M 0,1 = 1/P 01. and then show that for the nearly symmetric random walk above a i+1 M i,i+1 = a i + a i+1 + a i M i 1,i. 6
(d) [10 pts] Use the equation in part (c) to find an expression of M i,i+1 when a k = 1 (k + 1)(k + 2) for k 0. The expression for M i,i+1 cannot involve unevaluated summation. Then argue that to find M 0,n. Hint: 1 (k+1)(k+2) = 1 k+1 1 k+2. M 0,n = M 0,1 + M 1,2 + M 2,3 + + M n 1,n 7
Problem 4. [35 points] Suppose that people arrive at a bus stop in accordance with a Poisson process with rate λ. The bus departs at time t. (a) [5 pts] Suppose everyone arrived will wait until the bus comes, i.e., everyone arrive during [0, t] will get on the bus. What is the probability that the bus departs with n people aboard? (b) [10 pts] Let X be the total amount of waiting time of all those who get on the bus at time t. Find E[X]. (Hint: First find E[X N(t)] where N(t) is the number of people on the bus departs at time t.) 8
Suppose each person arrives at the bus stop will independently wait a period of time that has an exponential distribution with rate µ. If no bus arrives, he/she will leave the bus stop. (c) [10 pts] What is the probability that the bus departs with n people aboard? (d) [10 pts] If at time s (s < t), there are k people waiting at the bus stop. What is the expected number of customers who will get on the bus at time t? (Note some people may leave the bus stop and some may arrive.) 9