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1. Use this exercise to convince yourself that using different probabilities, the same discrete time chain may produce different stationary discrete time Markov chains with different transition matrices (we only consider two probabilities here in this problem; there are many other proba- bilities that can be chosen for which the process is not stationary or does not satisfy the Markov property). Consider two states 0 or 1 which a process (Xt) moves between. At each time step, t-0, 1, 2, 3, , a coin is flipped; at time t 0, if the coin is heads then the process starts in state 1, or if the coin is tails the process starts in state 0; for t 2 1, if the coin is heads, then the process stays in whichever state it is currently in, or if the coin is tails, then the process switches states (a) Suppose that we assign the underlying probability IP being one which would consider the coin to be fair and each flip independent. Give a brief justification why (X, 0 is a stationary discrete time Markov chain with respect to this probability P, and find the transition matrix P for the process (b) Suppose that we assign the underlying probability P being one which would consider the coin to be biased such that the probability of flipping a heads is 1/3; also suppose that with respect to this probability the coin flips are independent. Give a brief justification why (Xt)000 is a stationary discrete time Markov chain with respect to this probability P, and find the transition matrix P for the process.

The answer is one of the following:

Answers (in random order): 0.6,-2,-1,0, 1,2),5/36, 19/64,15/17.1/3 1-p p 1-p 0 1 00 0 1-p 0 114 0 3/4 1/21/2), 2/31/3). 0 1-p 0 p 0 1-p p 0 010

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math   Statistics-And-Probability   
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