Statistics 253/317 Introduction to Probability Models. Winter Midterm Exam Friday, Feb 8, 2013
|
|
- Cathleen Price
- 5 years ago
- Views:
Transcription
1 Statistics 253/317 Introduction to Probability Models Winter 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 TOTAL 100 1
2 Problem 1. [10 points] A Markov chain with state space {0, 1, 2, 3, 4} has transition matrix P = / / /2 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
3 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
4 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
5 (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
6 (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
7 (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
8 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
9 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
Statistics 253/317 Introduction to Probability Models. Winter Midterm Exam Monday, Feb 10, 2014
Statistics 253/317 Introduction to Probability Models Winter 2014 - Midterm Exam Monday, Feb 10, 2014 Student Name (print): (a) Do not sit directly next to another student. (b) This is a closed-book, closed-note
More informationStatistics 433 Practice Final Exam: Cover Sheet and Marking Sheet
Statistics 433 Practice Final Exam: Cover Sheet and Marking Sheet YOUR NAME INSTRUCTIONS: No notes, no calculators, and no communications devices are permitted. Please keep all materials away from your
More informationMarkov Chains. X(t) is a Markov Process if, for arbitrary times t 1 < t 2 <... < t k < t k+1. If X(t) is discrete-valued. If X(t) is continuous-valued
Markov Chains X(t) is a Markov Process if, for arbitrary times t 1 < t 2
More informationStochastic Processes
Stochastic Processes 8.445 MIT, fall 20 Mid Term Exam Solutions October 27, 20 Your Name: Alberto De Sole Exercise Max Grade Grade 5 5 2 5 5 3 5 5 4 5 5 5 5 5 6 5 5 Total 30 30 Problem :. True / False
More informationStatistics 150: Spring 2007
Statistics 150: Spring 2007 April 23, 2008 0-1 1 Limiting Probabilities If the discrete-time Markov chain with transition probabilities p ij is irreducible and positive recurrent; then the limiting probabilities
More informationIrreducibility. Irreducible. every state can be reached from every other state For any i,j, exist an m 0, such that. Absorbing state: p jj =1
Irreducibility Irreducible every state can be reached from every other state For any i,j, exist an m 0, such that i,j are communicate, if the above condition is valid Irreducible: all states are communicate
More informationStochastic process. X, a series of random variables indexed by t
Stochastic process X, a series of random variables indexed by t X={X(t), t 0} is a continuous time stochastic process X={X(t), t=0,1, } is a discrete time stochastic process X(t) is the state at time t,
More informationSTAT/MA 416 Midterm Exam 2 Thursday, October 18, Circle the section you are enrolled in:
STAT/MA 46 Midterm Exam 2 Thursday, October 8, 27 Name Purdue student ID ( digits) Circle the section you are enrolled in: STAT/MA 46-- STAT/MA 46-2- 9: AM :5 AM 3: PM 4:5 PM REC 4 UNIV 23. The testing
More informationRecap. Probability, stochastic processes, Markov chains. ELEC-C7210 Modeling and analysis of communication networks
Recap Probability, stochastic processes, Markov chains ELEC-C7210 Modeling and analysis of communication networks 1 Recap: Probability theory important distributions Discrete distributions Geometric distribution
More informationQuestion Points Score Total: 70
The University of British Columbia Final Examination - April 204 Mathematics 303 Dr. D. Brydges Time: 2.5 hours Last Name First Signature Student Number Special Instructions: Closed book exam, no calculators.
More informationSTA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008
Name STA 624 Practice Exam 2 Applied Stochastic Processes Spring, 2008 There are five questions on this test. DO use calculators if you need them. And then a miracle occurs is not a valid answer. There
More informationRandom Walk on a Graph
IOR 67: Stochastic Models I Second Midterm xam, hapters 3 & 4, November 2, 200 SOLUTIONS Justify your answers; show your work.. Random Walk on a raph (25 points) Random Walk on a raph 2 5 F B 3 3 2 Figure
More informationDO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO
QUESTION BOOKLET EE 26 Spring 2006 Final Exam Wednesday, May 7, 8am am DO NOT OPEN THIS QUESTION BOOKLET UNTIL YOU ARE TOLD TO DO SO You have 80 minutes to complete the final. The final consists of five
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran November 13, 2014.
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran November 13, 2014 Midterm Exam 2 Last name First name SID Rules. DO NOT open the exam until instructed
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Spring 2018 Kannan Ramchandran February 14, 2018.
EECS 126 Probability and Random Processes University of California, Berkeley: Spring 2018 Kannan Ramchandran February 14, 2018 Midterm 1 Last Name First Name SID You have 10 minutes to read the exam and
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Electrical Engineering and Computer Science 6.262 Discrete Stochastic Processes Midterm Quiz April 6, 2010 There are 5 questions, each with several parts.
More information8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains
8. Statistical Equilibrium and Classification of States: Discrete Time Markov Chains 8.1 Review 8.2 Statistical Equilibrium 8.3 Two-State Markov Chain 8.4 Existence of P ( ) 8.5 Classification of States
More informationSTAT 516 Midterm Exam 2 Friday, March 7, 2008
STAT 516 Midterm Exam 2 Friday, March 7, 2008 Name Purdue student ID (10 digits) 1. The testing booklet contains 8 questions. 2. Permitted Texas Instruments calculators: BA-35 BA II Plus BA II Plus Professional
More informationExam 2 Solutions. x 1 x. x 4 The generating function for the problem is the fourth power of this, (1 x). 4
Math 5366 Fall 015 Exam Solutions 1. (0 points) Find the appropriate generating function (in closed form) for each of the following problems. Do not find the coefficient of x n. (a) In how many ways can
More informationSTAT/MA 416 Midterm Exam 3 Monday, November 19, Circle the section you are enrolled in:
STAT/MA 46 Midterm Exam 3 Monday, November 9, 27 Name Purdue student ID ( digits) Circle the section you are enrolled in: STAT/MA 46-- STAT/MA 46-2- 9: AM :5 AM 3: PM 4:5 PM REC 4 UNIV 23. The testing
More informationIntroduction to Queuing Networks Solutions to Problem Sheet 3
Introduction to Queuing Networks Solutions to Problem Sheet 3 1. (a) The state space is the whole numbers {, 1, 2,...}. The transition rates are q i,i+1 λ for all i and q i, for all i 1 since, when a bus
More informationSOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012
SOLUTIONS IEOR 3106: Second Midterm Exam, Chapters 5-6, November 8, 2012 This exam is closed book. YOU NEED TO SHOW YOUR WORK. Honor Code: Students are expected to behave honorably, following the accepted
More informationDiscrete time Markov chains. Discrete Time Markov Chains, Limiting. Limiting Distribution and Classification. Regular Transition Probability Matrices
Discrete time Markov chains Discrete Time Markov Chains, Limiting Distribution and Classification DTU Informatics 02407 Stochastic Processes 3, September 9 207 Today: Discrete time Markov chains - invariant
More informationMARKOV PROCESSES. Valerio Di Valerio
MARKOV PROCESSES Valerio Di Valerio Stochastic Process Definition: a stochastic process is a collection of random variables {X(t)} indexed by time t T Each X(t) X is a random variable that satisfy some
More informationThe Transition Probability Function P ij (t)
The Transition Probability Function P ij (t) Consider a continuous time Markov chain {X(t), t 0}. We are interested in the probability that in t time units the process will be in state j, given that it
More informationBe sure this exam has 9 pages including the cover. The University of British Columbia
Be sure this exam has 9 pages including the cover The University of British Columbia Sessional Exams 2011 Term 2 Mathematics 303 Introduction to Stochastic Processes Dr. D. Brydges Last Name: First Name:
More informationHomework 4 Math 11, UCSD, Winter 2018 Due on Tuesday, 13th February
PID: Last Name, First Name: Section: Approximate time spent to complete this assignment: hour(s) Homework 4 Math 11, UCSD, Winter 2018 Due on Tuesday, 13th February Readings: Chapters 16.6-16.7 and the
More informationM.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics
MMT-008 Assignment Booklet M.Sc.(Mathematics with Applications in Computer Science) Probability and Statistics (Valid from 1 st July, 013 to 31 st May, 014) It is compulsory to submit the assignment before
More informationECE 302 Division 1 MWF 10:30-11:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding.
NAME: ECE 302 Division MWF 0:30-:20 (Prof. Pollak) Final Exam Solutions, 5/3/2004. Please read the instructions carefully before proceeding. If you are not in Prof. Pollak s section, you may not take this
More informationInterlude: Practice Final
8 POISSON PROCESS 08 Interlude: Practice Final This practice exam covers the material from the chapters 9 through 8. Give yourself 0 minutes to solve the six problems, which you may assume have equal point
More informationDisjointness and Additivity
Midterm 2: Format Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
More informationECE-517: Reinforcement Learning in Artificial Intelligence. Lecture 4: Discrete-Time Markov Chains
ECE-517: Reinforcement Learning in Artificial Intelligence Lecture 4: Discrete-Time Markov Chains September 1, 215 Dr. Itamar Arel College of Engineering Department of Electrical Engineering & Computer
More informationIE 5112 Final Exam 2010
IE 5112 Final Exam 2010 1. There are six cities in Kilroy County. The county must decide where to build fire stations. The county wants to build as few fire stations as possible while ensuring that there
More informationIEOR 6711, HMWK 5, Professor Sigman
IEOR 6711, HMWK 5, Professor Sigman 1. Semi-Markov processes: Consider an irreducible positive recurrent discrete-time Markov chain {X n } with transition matrix P (P i,j ), i, j S, and finite state space.
More informationMidterm 2 Review. CS70 Summer Lecture 6D. David Dinh 28 July UC Berkeley
Midterm 2 Review CS70 Summer 2016 - Lecture 6D David Dinh 28 July 2016 UC Berkeley Midterm 2: Format 8 questions, 190 points, 110 minutes (same as MT1). Two pages (one double-sided sheet) of handwritten
More informationCMPSCI 611 Advanced Algorithms Midterm Exam Fall 2015
NAME: CMPSCI 611 Advanced Algorithms Midterm Exam Fall 015 A. McGregor 1 October 015 DIRECTIONS: Do not turn over the page until you are told to do so. This is a closed book exam. No communicating with
More informationIE 336 Seat # Name (one point) < KEY > Closed book. Two pages of hand-written notes, front and back. No calculator. 60 minutes.
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.
More information6.041/6.431 Fall 2010 Final Exam Wednesday, December 15, 9:00AM - 12:00noon.
6.041/6.431 Fall 2010 Final Exam Wednesday, December 15, 9:00AM - 12:00noon. DO NOT TURN THIS PAGE OVER UNTIL YOU ARE TOLD TO DO SO Name: Recitation Instructor: TA: Question Score Out of 1.1 4 1.2 4 1.3
More informationComprehensive Exam in Real Analysis Fall 2006 Thursday September 14, :00-11:30am INSTRUCTIONS
Exam Packet # Comprehensive Exam in Real Analysis Fall 2006 Thursday September 14, 2006 9:00-11:30am Name (please print): Student ID: INSTRUCTIONS (1) The examination is divided into three sections to
More informationCS 798: Homework Assignment 3 (Queueing Theory)
1.0 Little s law Assigned: October 6, 009 Patients arriving to the emergency room at the Grand River Hospital have a mean waiting time of three hours. It has been found that, averaged over the period of
More informationDiscrete Event Systems Exam
Computer Engineering and Networks Laboratory TEC, NSG, DISCO HS 2016 Prof. L. Thiele, Prof. L. Vanbever, Prof. R. Wattenhofer Discrete Event Systems Exam Friday, 3 rd February 2017, 14:00 16:00. Do not
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More informationReadings: Finish Section 5.2
LECTURE 19 Readings: Finish Section 5.2 Lecture outline Markov Processes I Checkout counter example. Markov process: definition. -step transition probabilities. Classification of states. Example: Checkout
More informationExamination paper for TMA4265 Stochastic Processes
Department of Mathematical Sciences Examination paper for TMA4265 Stochastic Processes Academic contact during examination: Andrea Riebler Phone: 456 89 592 Examination date: December 14th, 2015 Examination
More informationEXAM IN COURSE TMA4265 STOCHASTIC PROCESSES Wednesday 7. August, 2013 Time: 9:00 13:00
Norges teknisk naturvitenskapelige universitet Institutt for matematiske fag Page 1 of 7 English Contact: Håkon Tjelmeland 48 22 18 96 EXAM IN COURSE TMA4265 STOCHASTIC PROCESSES Wednesday 7. August, 2013
More information(b) What is the variance of the time until the second customer arrives, starting empty, assuming that we measure time in minutes?
IEOR 3106: Introduction to Operations Research: Stochastic Models Fall 2006, Professor Whitt SOLUTIONS to Final Exam Chapters 4-7 and 10 in Ross, Tuesday, December 19, 4:10pm-7:00pm Open Book: but only
More informationUniversity of Illinois ECE 313: Final Exam Fall 2014
University of Illinois ECE 313: Final Exam Fall 2014 Monday, December 15, 2014, 7:00 p.m. 10:00 p.m. Sect. B, names A-O, 1013 ECE, names P-Z, 1015 ECE; Section C, names A-L, 1015 ECE; all others 112 Gregory
More informationIEOR 6711: Stochastic Models I, Fall 2003, Professor Whitt. Solutions to Final Exam: Thursday, December 18.
IEOR 6711: Stochastic Models I, Fall 23, Professor Whitt Solutions to Final Exam: Thursday, December 18. Below are six questions with several parts. Do as much as you can. Show your work. 1. Two-Pump Gas
More informationModelling data networks stochastic processes and Markov chains
Modelling data networks stochastic processes and Markov chains a 1, 3 1, 2 2, 2 b 0, 3 2, 3 u 1, 3 α 1, 6 c 0, 3 v 2, 2 β 1, 1 Richard G. Clegg (richard@richardclegg.org) November 2016 Available online
More informationNANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION MH4702/MAS446/MTH437 Probabilistic Methods in OR
NANYANG TECHNOLOGICAL UNIVERSITY SEMESTER I EXAMINATION 2013-201 MH702/MAS6/MTH37 Probabilistic Methods in OR December 2013 TIME ALLOWED: 2 HOURS INSTRUCTIONS TO CANDIDATES 1. This examination paper contains
More informationLecture 20 : Markov Chains
CSCI 3560 Probability and Computing Instructor: Bogdan Chlebus Lecture 0 : Markov Chains We consider stochastic processes. A process represents a system that evolves through incremental changes called
More informationIntro to Probability Instructor: Alexandre Bouchard
www.stat.ubc.ca/~bouchard/courses/stat302-sp2017-18/ Intro to Probability Instructor: Alexandre Bouchard Info on midterm CALCULATOR: only NON-programmable, NON-scientific, NON-graphing (and of course,
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This is a midterm from a previous semester. This means: This midterm contains problems that are of
More informationMATH 180A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM #2 FALL 2018
MATH 8A - INTRODUCTION TO PROBABILITY PRACTICE MIDTERM # FALL 8 Name (Last, First): Student ID: TA: SO AS TO NOT DISTURB OTHER STUDENTS, EVERY- ONE MUST STAY UNTIL THE EXAM IS COMPLETE. ANSWERS TO THE
More informationMathematics 24 Winter 2004 Exam I, January 27, 2004 In-class Portion YOUR NAME:
Mathematics 24 Winter 2004 Exam I, January 27, 2004 In-class Portion YOUR NAME: 1 1. Complete the following definitions: (a) Suppose that V is a vector space over a field F. Then W is a subspace of V if:
More informationSTOR : Lecture 17. Properties of Expectation - II Limit Theorems
STOR 435.001: Lecture 17 Properties of Expectation - II Limit Theorems Jan Hannig UNC Chapel Hill 1 / 14 Properties of expectation Recall: For two random variables X and Y, conditional distribution of
More informationUC Berkeley, CS 174: Combinatorics and Discrete Probability (Fall 2008) Midterm 1. October 7, 2008
UC Berkeley, CS 74: Combinatorics and Discrete Probability (Fall 2008) Midterm Instructor: Prof. Yun S. Song October 7, 2008 Your Name : Student ID# : Read these instructions carefully:. This is a closed-book
More informationMIDTERM Fundamental Algorithms, Spring 2008, Professor Yap March 10, 2008
INSTRUCTIONS: MIDTERM Fundamental Algorithms, Spring 2008, Professor Yap March 10, 2008 0. This is a closed book exam, with one 8 x11 (2-sided) cheat sheet. 1. Please answer ALL questions (there is ONE
More information(i) What does the term communicating class mean in terms of this chain? d i = gcd{n 1 : p ii (n) > 0}.
Paper 3, Section I 9H Let (X n ) n 0 be a Markov chain with state space S. 21 (i) What does it mean to say that (X n ) n 0 has the strong Markov property? Your answer should include the definition of the
More informationMarkov chains. Randomness and Computation. Markov chains. Markov processes
Markov chains Randomness and Computation or, Randomized Algorithms Mary Cryan School of Informatics University of Edinburgh Definition (Definition 7) A discrete-time stochastic process on the state space
More informationAssignment 3 with Reference Solutions
Assignment 3 with Reference Solutions Exercise 3.: Poisson Process Given are k independent sources s i of jobs as shown in the figure below. The interarrival time between jobs for each source is exponentially
More information1 Continuous-time chains, finite state space
Université Paris Diderot 208 Markov chains Exercises 3 Continuous-time chains, finite state space Exercise Consider a continuous-time taking values in {, 2, 3}, with generator 2 2. 2 2 0. Draw the diagramm
More informationIEOR 6711: Stochastic Models I SOLUTIONS to the First Midterm Exam, October 7, 2008
IEOR 6711: Stochastic Models I SOLUTIONS to the First Midterm Exam, October 7, 2008 Justify your answers; show your work. 1. A sequence of Events. (10 points) Let {B n : n 1} be a sequence of events in
More informationMATH 56A: STOCHASTIC PROCESSES CHAPTER 2
MATH 56A: STOCHASTIC PROCESSES CHAPTER 2 2. Countable Markov Chains I started Chapter 2 which talks about Markov chains with a countably infinite number of states. I did my favorite example which is on
More informationOutlines. Discrete Time Markov Chain (DTMC) Continuous Time Markov Chain (CTMC)
Markov Chains (2) Outlines Discrete Time Markov Chain (DTMC) Continuous Time Markov Chain (CTMC) 2 pj ( n) denotes the pmf of the random variable p ( n) P( X j) j We will only be concerned with homogenous
More informationStatistics & Data Sciences: First Year Prelim Exam May 2018
Statistics & Data Sciences: First Year Prelim Exam May 2018 Instructions: 1. Do not turn this page until instructed to do so. 2. Start each new question on a new sheet of paper. 3. This is a closed book
More informationMAT SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions.
MAT 4371 - SYS 5120 (Winter 2012) Assignment 5 (not to be submitted) There are 4 questions. Question 1: Consider the following generator for a continuous time Markov chain. 4 1 3 Q = 2 5 3 5 2 7 (a) Give
More informationLTCC. Exercises. (1) Two possible weather conditions on any day: {rainy, sunny} (2) Tomorrow s weather depends only on today s weather
1. Markov chain LTCC. Exercises Let X 0, X 1, X 2,... be a Markov chain with state space {1, 2, 3, 4} and transition matrix 1/2 1/2 0 0 P = 0 1/2 1/3 1/6. 0 0 0 1 (a) What happens if the chain starts in
More information16:330:543 Communication Networks I Midterm Exam November 7, 2005
l l l l l l l l 1 3 np n = ρ 1 ρ = λ µ λ. n= T = E[N] = 1 λ µ λ = 1 µ 1. 16:33:543 Communication Networks I Midterm Exam November 7, 5 You have 16 minutes to complete this four problem exam. If you know
More informationQ = (c) Assuming that Ricoh has been working continuously for 7 days, what is the probability that it will remain working at least 8 more days?
IEOR 4106: Introduction to Operations Research: Stochastic Models Spring 2005, Professor Whitt, Second Midterm Exam Chapters 5-6 in Ross, Thursday, March 31, 11:00am-1:00pm Open Book: but only the Ross
More informationM.Sc. (MATHEMATICS WITH APPLICATIONS IN COMPUTER SCIENCE) M.Sc. (MACS)
No. of Printed Pages : 6 MMT-008 M.Sc. (MATHEMATICS WITH APPLICATIONS IN COMPUTER SCIENCE) M.Sc. (MACS) Term-End Examination 0064 December, 202 MMT-008 : PROBABILITY AND STATISTICS Time : 3 hours Maximum
More information- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,
More information(Practice Version) Midterm Exam 2
EECS 126 Probability and Random Processes University of California, Berkeley: Fall 2014 Kannan Ramchandran November 7, 2014 (Practice Version) Midterm Exam 2 Last name First name SID Rules. DO NOT open
More informationNotes on Continuous Random Variables
Notes on Continuous Random Variables Continuous random variables are random quantities that are measured on a continuous scale. They can usually take on any value over some interval, which distinguishes
More informationMath 365 Final Exam Review Sheet. The final exam is Wednesday March 18 from 10am - 12 noon in MNB 110.
Math 365 Final Exam Review Sheet The final exam is Wednesday March 18 from 10am - 12 noon in MNB 110. The final is comprehensive and will cover Chapters 1, 2, 3, 4.1, 4.2, 5.2, and 5.3. You may use your
More information1 Basic continuous random variable problems
Name M362K Final Here are problems concerning material from Chapters 5 and 6. To review the other chapters, look over previous practice sheets for the two exams, previous quizzes, previous homeworks and
More informationUNIVERSITY OF LONDON IMPERIAL COLLEGE LONDON
UNIVERSITY OF LONDON IMPERIAL COLLEGE LONDON BSc and MSci EXAMINATIONS (MATHEMATICS) MAY JUNE 23 This paper is also taken for the relevant examination for the Associateship. M3S4/M4S4 (SOLUTIONS) APPLIED
More information1 Gambler s Ruin Problem
1 Gambler s Ruin Problem Consider a gambler who starts with an initial fortune of $1 and then on each successive gamble either wins $1 or loses $1 independent of the past with probabilities p and q = 1
More informationCSE 312 Foundations, II Final Exam
CSE 312 Foundations, II Final Exam 1 Anna Karlin June 11, 2014 DIRECTIONS: Closed book, closed notes except for one 8.5 11 sheet. Time limit 110 minutes. Calculators allowed. Grading will emphasize problem
More informationSTAT FINAL EXAM
STAT101 2013 FINAL EXAM This exam is 2 hours long. It is closed book but you can use an A-4 size cheat sheet. There are 10 questions. Questions are not of equal weight. You may need a calculator for some
More informationSTAT STOCHASTIC PROCESSES. Contents
STAT 3911 - STOCHASTIC PROCESSES ANDREW TULLOCH Contents 1. Stochastic Processes 2 2. Classification of states 2 3. Limit theorems for Markov chains 4 4. First step analysis 5 5. Branching processes 5
More informationExercises Stochastic Performance Modelling. Hamilton Institute, Summer 2010
Exercises Stochastic Performance Modelling Hamilton Institute, Summer Instruction Exercise Let X be a non-negative random variable with E[X ]
More informationP i [B k ] = lim. n=1 p(n) ii <. n=1. V i :=
2.7. Recurrence and transience Consider a Markov chain {X n : n N 0 } on state space E with transition matrix P. Definition 2.7.1. A state i E is called recurrent if P i [X n = i for infinitely many n]
More informationMath 51 Midterm 1 July 6, 2016
Math 51 Midterm 1 July 6, 2016 Name: SUID#: Circle your section: Section 01 Section 02 (1:30-2:50PM) (3:00-4:20PM) Complete the following problems. In order to receive full credit, please show all of your
More informationTMA4265 Stochastic processes ST2101 Stochastic simulation and modelling
Norwegian University of Science and Technology Department of Mathematical Sciences Page of 7 English Contact during examination: Øyvind Bakke Telephone: 73 9 8 26, 99 4 673 TMA426 Stochastic processes
More informationAt the boundary states, we take the same rules except we forbid leaving the state space, so,.
Birth-death chains Monday, October 19, 2015 2:22 PM Example: Birth-Death Chain State space From any state we allow the following transitions: with probability (birth) with probability (death) with probability
More informationSolution: By Markov inequality: P (X > 100) 0.8. By Chebyshev s inequality: P (X > 100) P ( X 80 > 20) 100/20 2 = The second estimate is better.
MA 485-1E, Probability (Dr Chernov) Final Exam Wed, Dec 12, 2001 Student s name Be sure to show all your work. Each problem is 4 points. Full credit will be given for 9 problems (36 points). You are welcome
More informationExample: physical systems. If the state space. Example: speech recognition. Context can be. Example: epidemics. Suppose each infected
4. Markov Chains A discrete time process {X n,n = 0,1,2,...} with discrete state space X n {0,1,2,...} is a Markov chain if it has the Markov property: P[X n+1 =j X n =i,x n 1 =i n 1,...,X 0 =i 0 ] = P[X
More information1. If X has density. cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. f(x) =
1. If X has density f(x) = { cx 3 e x ), 0 x < 0, otherwise. Find the value of c that makes f a probability density. 2. Let X have density f(x) = { xe x, 0 < x < 0, otherwise. (a) Find P (X > 2). (b) Find
More informationEXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY
EXAMINATIONS OF THE ROYAL STATISTICAL SOCIETY GRADUATE DIPLOMA, 011 MODULE 3 : Stochastic processes and time series Time allowed: Three Hours Candidates should answer FIVE questions. All questions carry
More informationThis exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner.
GROUND RULES: This exam contains 6 questions. The questions are of equal weight. Print your name at the top of this page in the upper right hand corner. This exam is closed book and closed notes. Show
More informationModelling data networks stochastic processes and Markov chains
Modelling data networks stochastic processes and Markov chains a 1, 3 1, 2 2, 2 b 0, 3 2, 3 u 1, 3 α 1, 6 c 0, 3 v 2, 2 β 1, 1 Richard G. Clegg (richard@richardclegg.org) December 2011 Available online
More informationEECS 126 Probability and Random Processes University of California, Berkeley: Spring 2018 Kannan Ramchandran February 14, 2018.
EECS 6 Probability and Random Processes University of California, Berkeley: Spring 08 Kannan Ramchandran February 4, 08 Midterm Last Name First Name SID You have 0 minutes to read the exam and 90 minutes
More informationWinter 2014 Practice Final 3/21/14 Student ID
Math 4C Winter 2014 Practice Final 3/21/14 Name (Print): Student ID This exam contains 5 pages (including this cover page) and 20 problems. Check to see if any pages are missing. Enter all requested information
More informationLecture 20: Reversible Processes and Queues
Lecture 20: Reversible Processes and Queues 1 Examples of reversible processes 11 Birth-death processes We define two non-negative sequences birth and death rates denoted by {λ n : n N 0 } and {µ n : n
More informationLecture 11: Introduction to Markov Chains. Copyright G. Caire (Sample Lectures) 321
Lecture 11: Introduction to Markov Chains Copyright G. Caire (Sample Lectures) 321 Discrete-time random processes A sequence of RVs indexed by a variable n 2 {0, 1, 2,...} forms a discretetime random process
More informationMidterm 2. Your Exam Room: Name of Person Sitting on Your Left: Name of Person Sitting on Your Right: Name of Person Sitting in Front of You:
CS70 Discrete Mathematics and Probability Theory, Fall 2018 Midterm 2 8:00-10:00pm, 31 October Your First Name: SIGN Your Name: Your Last Name: Your SID Number: Your Exam Room: Name of Person Sitting on
More informationNAME : Math 20. Final Exam August 27, Prof. Pantone
NAME : Math 0 Final Exam August 7, 07 Prof. Pantone Instructions: This is a closed book exam and no notes are allowed. You are not to provide or receive help from any outside source during the exam except
More informationPositive and null recurrent-branching Process
December 15, 2011 In last discussion we studied the transience and recurrence of Markov chains There are 2 other closely related issues about Markov chains that we address Is there an invariant distribution?
More informationP(X 0 = j 0,... X nk = j k )
Introduction to Probability Example Sheet 3 - Michaelmas 2006 Michael Tehranchi Problem. Let (X n ) n 0 be a homogeneous Markov chain on S with transition matrix P. Given a k N, let Z n = X kn. Prove that
More information