Emily Winn -- Markov chains and their applications to stochastic processes in the 
  discrete time case

  Generally excellent job on the paper and the presentation!

  
  A few specific comments:

  1)  p. 3 -- "The goal is to determine what the weather in Oz will be based on today’s 
      weather, be it rainy, nice, or snowy."  I think it's more accurate to say this is a 
      model that is based on the probabilities.  It's not that you're trying to "determine"
      what the weather will be the next day; it's that you are modeling the probabilities
      given the weather the previous day.

  2)  Your proof of Theorem 1 is fine, but a nicer (more "elegant") way to do that would 
      be to use a proof by mathematical induction :)

  3)  p. 7 -- Stray partial sentence after the statement of Theorem 2.

  4)  p. 7.  The discussion of the Strong Markov Property is a bit incomplete because
      you have not really explained what a stopping time is.  You say a stopping 
      time is "the random instant determined from the observation of a Markov chain and 
      thus the state can be taken without knowledge of future states."  But to make 
      sense of the proof, I think you really need to say more about the formal definition:
      a stopping time is another random variable $T : \Omega \to {\bf N}$ with the property 
      that the event $T = n$ depends only on $X_0, \ldots, X_n$.  

  5)  p. 12. A comment: The proof of the Limit Theorem is very similar to the idea of the 
      "power method" for determining the dominant eigenvalue of a matrix as studied
      in numerical analysis.

  6)  p. 15.  Instead of saying the Wright-Fisher model says individuals can mate sexually
      and asexually, I think it might be better just to say that the Wright-Fisher model 
      doesn't incorporate a gender at all for individuals and all pairs can "mate".  

  7)  p. 18. "Markov chains support the idea of quarantine" -- that's true, but
      virtually any epidemic model does the same (e.g. deterministic ODE models
      show that quarantine is effective too).  

  Final Project Grade Computation

  Bibliography:  10/10

  Paper:         58/60

  Presentation:  30/30

  Total:         98/100