INE 312 • Final • Operations Research II

Random Variables and Probability Distributions

Conditional Probability

Total Probability Rule

Expected Value

Review Question 1

Review Question 2

Review Question 3

What is a Markov Chain?

One-Step Transition Probabilities

Example

n-Step Transition Probabilities

Chapman-Kolmogorov Equations

Example

Unconditional State Probabilities

Example

Steady State Distribution

Example

Classes and State Properties

Periodicity and Ergodic Markov Chains

Example

Example

Example

First Passage Times

Example

Expected First Passage Time

Expected Recurrence Time

Example

Absorbing Markov Chains

Matrix Structure

Inverse of a Matrix

Expected Number of State Visits

Example

Expected Time Until Absoption

Example

Absorption Probabilities

Example

Markov Chains 1

Markov Chains 2

Markov Chains 3

Markov Chains 4

Classification of States and Limiting Distribution 1

Classification of States and Limiting Distribution 2

Classification of States and Limiting Distribution 3

Classification of States and Limiting Distribution 4

Classification of States and Limiting Distribution 5

Classification of States and Limiting Distribution 6

Classification of States and Limiting Distribution 7

Mean First Passage Times 1

Mean First Passage Times 2

Absorbing Markov Chains 1

Absorbing Markov Chains 2