Markov Chains

Description Sometimes we are interested in how a random variable changes over time. The study of how a random variable evolves over time includes stochastic processes. An explanation of stochastic processes – in particular, a type of stochastic process known as a Markov chain is included. We begin by defining the concept of a stochastic process.

5.1 What is a Stochastic Process? ,[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object]

5.2 What is a Markov Chain? ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object]

The Gambler’s Ruin Problem ,[object Object],[object Object],[object Object]

The Gambler’s Ruin Problem ,[object Object]

5.3 n -Step Transition Probabilities ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Cola Example ,[object Object],[object Object],[object Object],[object Object],[object Object]

The Cola Example ,[object Object],[object Object]

The Cola Example ,[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

5.4 Classification of States in a Markov Chain ,[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

5.5 Steady-State Probabilities and Mean First Passage Times ,[object Object],[object Object]

Transient Analysis & Intuitive Interpretation ,[object Object],[object Object],[object Object]

Steady-State Probabilities ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

An Intuitive Interpretation of Steady-State Probabilities ,[object Object],[object Object],[object Object]

Use of Steady-State Probabilities in Decision Making ,[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],[object Object],[object Object],[object Object],[object Object]

Inventory Example ,[object Object],[object Object],[object Object],[object Object],[object Object]

Inventory Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Inventory Example: (One-Step) Transition Matrix ,[object Object],[object Object],[object Object],[object Object]

Inventory Example: Transition Diagram 0 1 2 3

Inventory Example: (One-Step) Transition Matrix

Transition Matrix: Two-Step ,[object Object]

Transition Matrix: Four-Step ,[object Object]

Transition Matrix: Eight-Step ,[object Object]

Steady-State Probabilities ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Steady-State Probabilities: Inventory Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Mean First Passage Times ,[object Object],[object Object]

Solving for Steady-State Probabilities and Mean First Passage Times on the Computer ,[object Object],[object Object]

5.6 Absorbing Chains ,[object Object],[object Object],[object Object]

Accounts Receivable Example ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],New 1 month 2 months 3 months Paid Bad Debt New 1 month 2 months 3 months Paid Bad Debt

[object Object],[object Object],[object Object],m columns s-m rows m rows s-m columns P =

[object Object],[object Object],New 1 month 2 months 3 months Paid Bad Debt New 1 month 2 months 3 months Paid Bad Debt Q R

Markov Chains

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Markov Chains