MATH 303

Syllabus: pdf

Outline:

  • Jan 6: Motivation (slides)/ Discrete Time Markov chains (Markov property, transition matrix, notes)
  • Jan 8: Stationary distribution, transition diagram, Chapman-Kolmogorov equations (notes)
  • Jan 10: Classification of states: accessibility, communicating classes, period (notes)
  • Jan 13: Transient and recurrent states (notes)
  • Jan 17: Transient and recurrent states, Gambler’s ruin problem (notes)
  • Jan 20: Gambler’s ruin problem, Recurrence of random walk on Z (notes)
    – Some references on difference equations:  for the method with order 2, you can follow this link; to go deeper: classical reference (book), or this link, which makes the connection with linear algebra.
    – On the Stirling formula: see this link for a derivation, or for more complete expansion, here
  • Jan 22: Recurrence of random walk on Z^d (notes)
  • Jan 24: Recurrence of random walk on Z^d (notes)
  • Jan 27: Limiting Probabilities (notes)
  • Jan 29: Limiting Probabilities, doubly stochastic matrix, Time reversibility (notes)
  • Jan 31: Time reversibility (notes)
  • Feb 3: Time reversibility (Kolmogorov’s criterion), Ehrenfest chain (notes)
  • Feb 5: Ehrenfest chain, branching process (notes)
  • Feb 7: Branching process, generating function (notes)
  • Feb 10: Branching process (notes)
  • Feb 12: Poisson Process (Introduction, Exponential Distribution) (notes)
  • Feb 14: Exponential distribution and no memory property (notes)
  • Feb 24: Minimum and sum of exponential r.v.’s (notes)
  • Feb 26: Sum of exponential r.v.’s, first description of Poisson Process (notes)
  • Feb 28: Poisson Process (pmf, stationary and increment properties, second definition) (notes)
  • Mar 2: Poisson Process (equivalency of the two definitions) (notes)
  • Mar 4: Superposition and thinning of Poisson process (notes)
  • Mar 6: Conditional arrival times (notes)
  • Mar 9: Conditional arrival times, sampling (notes)
  • Mar 11: Sampling, Continuous time Markov chain (notes)
  • Mar 13: Birth death process (notes)
  • Mar 16: From now on we move to online course (see Canvas)
  • April 3: Special lecture on modeling of COVID-19: deterministic modeling / stochastic modeling (~27.MB, to watch video at the end, open with acrobat reader)

Homework:

Midterm 1:

Midterm 2:

  • Cancelled (see Canvas)