3.5 Markov Chains - The Classifications of States - Exercises.pdf - THE CHINESE UNIVERSITY OF HONG KONG Department of Statistics STAT3007 Introduction

# 3.5 Markov Chains - The Classifications of States - Exercises.pdf

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THE CHINESE UNIVERSITY OF HONG KONG Department of Statistics STAT3007: Introduction to Stochastic Processes Markov Chains - The Classification of States - Exercises 1. (Exercise 4.3.1 in Pinsky and Karlin) A Markov chain has a transition probability matrix 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 . 5 0 0 0 0 0 . 5 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 Find the equivalence classes. 2. (Exercise 4.3.2 in Pinsky and Karlin) A Markov chain has a transition probability matrix 1 / 3 0 1 / 3 0 0 1 / 3 1 / 2 1 / 4 1 / 4 0 0 0 0 0 0 0 1 0 1 / 4 1 / 4 1 / 4 0 0 1 / 4 0 0 1 0 0 0 0 0 0 0 0 1 Find the communicating classes. 3. (Exercise 4.3.3a in Pinsky and Karlin) A Markov chain on states { 0 , 1 , 2 , 3 , 4 , 5 } has transition probability matrix 1 / 3 0 2 / 3 0 0 0 0 1 / 4 0 3 / 4 0 0 2 / 3 0 1 / 3 0 0 0 0 1 / 5 0 4 / 5 0 0 1 / 4 1 / 4 0 0 1 / 4 1 / 4 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 1 / 6 Find all communicating classes.

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