# Markov Chains and Applications of Matrix Methods: Fixed Point and Absorbing Markov Chains (UMAP)

### Author: Sister Mary K. Keller

The Markov process pulls together related prior probabilities into chains of events. The description of the process in matrix form allows one to make long-range predictions. Students define a Markov chain, interpret powers of matrices representing Markov chains, recognize certain processes as Markov chains, formulate a matrix of transition probabilties from a tree diagram of a Markov chain.** Table of Contents:1. MARKOV CHAINS** 1.1 Introduction

1.2 Tree Diagrams

1.3 Calculating Probabilities From a Tree Diagram

1.4 The Matrix Representation of a Markov Chain

1.5 Experiment 1

1.6 Experiment 2

1.7 Model Exam (Unit 107)

**2. APPLICATIONS OF MATRIX METHODS: FIXED POINT AND ABSORBING MARKOV CHAINS**

2.1 Challenge Problem

2.2 Regular Transition Matrices

2.3 Fixed-Probability Vectors

2.4 Calculating a Fixed-Probability Vector

2.5 Experiment 1

2.6 A Fixed Probability Vector from a System of Linear Functions

2.7 Experiment 2

2.8 Experiment 3

2.9 Absorbing Markov Chains

2.10 A Second Challenge Problem

2.11 Standard Form for an Absorbing Markov Chain

2.12 Partitioning the Standard Form

2.13 Making Decision Based on Probability

2.14 The Probability of Reaching a Given Absorbing State

2.15 Experiment 4

2.16 Model Exam (Unit 111)

**3. ANSWERS TO EXERCISES (UNIT 107)**

4. ANSWERS TO MODEL EXAM (UNIT 107)

5. ANSWERS TO MODEL EXAM (UNIT 111)

APPENDIX A

4. ANSWERS TO MODEL EXAM (UNIT 107)

5. ANSWERS TO MODEL EXAM (UNIT 111)

APPENDIX A

#### Mathematics Topics:

#### Application Areas:

#### Prerequisites:

You must have a **Full Membership** to download this resource.

If you're already a member, **login here**.