PhD Qualifying Exam Information for Advanced Stochastic Processes

Rules

  1. Students have four hours to complete the exam and submit their exam materials.
  2. The exam is open book and open notes.
  3. Students are allowed to use a calculator (not a computer) during the exam.
  4. Students may not communicate with others during the exam.

Exam Integrity

  1. 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.
  2. When submitting their exam materials, students shall turn in all written work created during the exam.
  3. 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