Math 331 - Introduction to Probability and Markov Chain modelling

• Prerequisites: Math 234 or Math 222 and 240
• Frequency: Fall (I), Spring (II)
• Student Body: Indicated for Computer Science majors
• Credits: 3. (N-A)
• Recent Texts: "Fundamentals of Probability" by Saeed Ghahramani.
• Course Coordinator: Jim Kuelbs
• Background and Goals: Math 331 concentrates on discrete models in probability, and betond basic introduction to the subject, it presents material on Markov chains. It is similar to Math 431, but our focus on discrete models allows us to go a bit further into the subject. Designed for Computer Science majors.
• Alternatives: Math 431 also includes continuous models
• Subsequent Courses: N/A

Content coverage:

• The probability model (9-10 hours)
  • sample spaces and events
  • probabilities of events
  • conditional probabilities
  • independent events
  • Bayes formula
• Random Variables 15-18 hours
  • discrete random variables with a discussion of the binomial, geometric, and Poisson random variables
  • continuous random variables with a discussion of the normal random variable
  • expectation and variancs: Markov's and Chebyshev's inequality with some examples. Also Chernoff's inequality if time permits
  • jointly distributed random variables and independence
  • sums of independent random variables
  • the law of large numbers and the central limit theorem (including confidence intervals with some discussion of testing for the mean of a distribution)
• Random walks (3-5 hours, to motivate the Markov chain model)
  • simple random walks and the gambler's ruin problem i-the probability of ruin, ii-the expected duration of the game
  • recurrence and transience of a simple random walk
• Markov Chains in discrete time and space (10-12 hours)
  • the transition function and the initial distribution. the Chapman-Kolmogorov equations
  • recurrence and trancience
  • classes of states
  • absorption probabilities
  • limiting transition probabilities, stationary distributions
  • examples