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Markov Chain: Invariant Distribution

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Please see attached file.

Consider the Markov chain with transition matrix (please see the attached file) where 0 <= p <= 1 and 0 <= q <= 1. Find an invariant distribution of this chain, For which values of p and q is the distribution unique? Given an initial distribution (please see the attached file), for which values of p and q will this distribution evolve toward the unique invariant distribution?

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This solution provides a detailed, step-by-step explanation of how to solve the given problems involving a given Markov chain.

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