Department of Industrial Engineering
4207 Bell Engineering Center
1 University of Arkansas
Fayetteville, AR 72701
Phone: (479) 575-3156
Fax: (479) 575-8431
PhD Qualifying Exam Information for Advanced Stochastic Processes
Rules
- Students have four hours to complete the exam and submit their exam materials.
- The exam is open book and open notes.
- Students are allowed to use a calculator (not a computer) during the exam.
- Students may not communicate with others during the exam.
Exam Integrity
- Within 24 hours of receiving confirmation that the exam materials have been received, students shall delete all files created during the exam and empty the trash/recycle bin on any devices used during the exam.
- When submitting their exam materials, students shall turn in all written work created during the exam.
- After completing the exam, students shall not communicate any information about the exam to anyone other than a member of the INEG faculty.
Topics
Probability Theory
- Fundamentals of probability theory
- conditional probability
- conditional expectation
- Inequalities and limit theorems
- Transforms (generating functions)
- probability generating function
- moment generating function
- characteristic function
Poisson Processes
- Definitions of Poisson process
- The memoryless property
- Interarrival and waiting time distributions
- Conditional distribution of the arrival times
- Sampling a Poisson process
- Nonhomogeneous Poisson processes
- Compound Poisson processes
Renewal Theory
- Renewal function and propositions of renewal function
- Limit theorems (including elementary renewal theorems and the key renewal theorem)
- Renewal reward process
- The Formula of Little
- PASTA property
- The Pollaczek–Khintchine Formula
Discrete-time Markov Chain
- Random walk and related propositions
- Chapman-Kolmogorov equations
- Classification of states; transitions among states; mean time spent in transient states
- Limit theorems; stationary and limiting probabilities; transient behavior
Continuous-time Markov Chain
- Birth and death process
- Transition probability function; Kolmogorov backward and forward equations
- Limiting probabilities, transient distributions
- The uniformization method