MATH670: Advanced Statistical Theory

Course Description

Objectives

1. Develop an understanding of the underlying theory of probability, random variables, probability distributions, and the behavior of stochastic processes and Markov chains.
   
   
2. Show how matrices are applied to modern problems of estimating the behavior of probabilistic systems.

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Grading Procedure

Attendance Policy

Course Outline

• Chapters 1 - 3: Reviewed as needed (6 days)

  • Chapter 1: Introduction to Probability Theory
  • Chapter 2: Random Variables
  • Chapter 3: Conditional Probability and Expectation
  
  
  

• Chapter 4: Markov Chains (9 days)

  Introduction, Chapman-Kolmogorov equations, classification of states, limiting probabilities, applications, branching processes, time reversible Markov chains, Markov decision processes
  
  

• Chapter 5: The Exponential Distribution and the Poisson Process (9 days)

  Introduction, exponential distribution, properties of the exponential distribution, poisson process, waiting time, properties of the poisson process, conditional distribution of arrival time, generalizations of the poisson process
  
  

• Chapter 6: Continuous Time Markov Chains (9 days)

  Continuous time Markov chains, birth and death processes, Kolmogorov difference equations, limiting probabilities, time reversibility, uniformization, computing transition probabilities
  
  

• Chapter 7: Renewal Theory and Its Application (9 days)

  Distribution of N(t), limit theorems and their applications, renewal reward processes, regenerative processes, semi-Markov processes, the inspection paradox, computing renewal function
  
  
• Note: At appropriate places in this course, time should be allotted to elaborate on the historical aspects relevant to the subject.