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1. (Wheel of Fortune Continued). Consider again the Markov chains corresponding to the wheel of fortune. Prove or disprove that they are irreducible and aperiodic.

2. * (Stationary Distribution in Ehrenfest Model). Consider the general Ehrenfest chain defined in the text, with m balls, and transfer probabilities ?; ?; 0 ?; ? 1. Identify a stationary distribution if it exists.

3. * (Time Until Break away). Consider a general stationary Markov chain fXng, and let T D minfn > 1W Xn ¤ X0g.

(a) Can T be equal to 1 with a positive probability?

(b) Give a simple necessary and sufficient condition for P.T 1/ D 1.

(c) For the weather pattern, Ehrenfest urn, and the cat and mouse chain, compute E.T jX0 D i/ for a general i in the corresponding state space S.

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  • Category:- Statistics and Probability
  • Reference No.:- M91641488

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