HOMEWORK.

Assignment 1: due Wednesday, April 6, 2005, in class.
Please hand in one page answering the following questions.
A. In one paragraph, describe your background and interest in taking this course. Indicate any specific topics you are interested in.
B. In one paragraph,
(a) if you are from a department other than Mathematics, describe a problem where you would like to be able to use a stochastic process model; then try to describe in words what X(t), the state of the process at time t would represent. Specify the time index set and the state space for your process. Try to formulate questions that you would like to be able to answer for this process. Try to draw a typical sample path.
(b) for Mathematics students, or those who do not have a problem in mind, describe a phenomenon in the real world where you think a stochastic process model would be useful and then answer the same questions as in (a) above.
C. Describe the state space for a stochastic process that describes the random evolution in time of a finite DNA segment, where mutations, insertions and deletions of sites may occur. (You may assume that there are 4 possible values at each site, A, C, G, T.)

Assignment 2, due Monday, April 25, 2005, in class.
Problems from the text by Lawler: 1.1, 1.3, 1.5, 1.6, 1.8, 1.11.
In addition, do the following problem. Let {S(n):n=0,1,2, ...} be a simple symmetric random walk on the integers that starts at zero and at each step goes up by one, down by one, or stays where it is, with equal probability (probability 1/3 for each possibility). Consider the process Z(n) = max{S(0), S(1), ... , S(n)} for n=0,1,2, .... Show that {Z(n), n=0,1, 2, ...} is NOT a Markov chain.

Assignment 3, due Wednesday, May 11, 2005, in class.
Problems from the text by Lawler: 2.2, 7.1.
Problem A: Consider a Hidden Markov Model representation of a coin tossing experiment. Assume a two-state model (corresponding to two coins) with probability of output (of H or T) given by: P(H|Coin 1) = 0.5, P(T|Coin 1) = 0.5 and P(H|Coin 2)= 0.25, P(T|Coin 2)=0.75. Assume that the transition probability matrix for the Markov chain governing the coin state is: P(11) = 0.7, P(12)=0.3, P(21)=0.4, P(22) =0.6. where P(ij) denotes the probability that given that the coin is currently in state i, it is in state j at the next time step. Assume that initially the coin is equally likely to be coin 1 or 2.
(a) Suppose that you observe the sequence HHTTT. What is a/the most likely state sequence to generate this output?
(b) What is the probability that given the observation sequence it was generated entirely by coin 2?
Problem B: For the profile HMM given out in class, compute the probability of occurence of the path and output shown (the output is ACCY). That is, if O is the output sequence and Q is the associated sequence of hidden Markov model states, compute P(O, Q). Note that the transition probability from the last insert state to End should be 0.52 rather than 1.

Assignment 4, due Wednesday, June 1, 2005, in class.
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Assignment 5, due Wednesday, June 8, 2005, 5pm. (OPTIONAL)
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