An Introduction to Stochastic Modeling

Abstract

Stochastic Modeling. Probability Review. The Major Discrete Distributions. @rtant Continuous Distributions. Some Elementary Exercises. Useful Functions, Integrals, and Sums. Conditional Probability and Conditional Expectation: The Discrete Case. The Dice Game Craps. Random Sums. Conditioning on a Continuous Random Variable. Markov Chains: Introduction: Definitions. Transition Probability Matrices of a Markov Chain. Some Markov Chain Models. First Step Analysis. Some Special Markov Chains. Functionals of Random Walks and Success Runs. Another Look at First Step Analysis. The Long Run Behavior of Markov Chains: Regular Transition Probability Matrices. Examples. The Classification of States. The Basic Limit Theorem of Markov Chains. Reducible Markov Chains. Sequential Decisions and Markov Chains. Poisson Processes: The Poisson Distribution and the Poisson Processes. The Law of Rare Events. Distributions Associated with the Poisson Process. The Uniform Distribution and Poisson Processes. Spatial Poisson Processes. Compound and Marked Poisson Processes. Continuous Time Markov Chains: Pure Birth Processes. Ptire Death Processes. Birth and Death Processes. The Limiting Behavior of Birth and Death Processes. Birth and Death Processes with Absorbing States. Finite State Continuous Time Markov Chains. Set Valued Processes. Renewal Phenomena: Definition of a Renewal Process and Related Concepts. Some Examples of Renewal Processes. The Poisson Process Viewed as a Renewal Process. The Asymptotic ]3ehavior as Renewal Process.