STAT_333_Assignment_3

STAT_333_Assignment_3 - STAT 333 Assignment 3 Due Monday...

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STAT 333 Assignment 3 Due: Monday, April 5 at the beginning of class 1. Consider a sequence of repeated independent tosses of a fair coin, each toss resulting in H or T. For each n = 1, 2, 3, . . . define X n = length of the current run including the n th toss where a run is a sequence of all the same outcomes (i.e., all H or all T). For example, if the sequence of outcomes looks like H H T H H H H T … then X 1 = 1, X 2 = 2, X 3 = 1, X 4 = 1, X 5 = 2, X 6 = 3, X 7 = 4, X 8 = 1, etc. a. Model this as a Markov chain by writing down the state space S and transition matrix P . b. Prove that this chain is irreducible and find the period of the chain. c. Prove that this chain is positive recurrent by solving recursively for the unique equilibrium distribution π = (π 1 , π 2 , π 3 , …). What distribution is this? What is the expected number of tosses between returns to state 4? d. Suppose the coin is not fair. Why can this not be modelled directly as a Markov chain as above? What could you do (how could you enlarge the state space) to enable you to model this as a Markov chain?

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