ADVANCED PROBABILITY and APLICATIONS

AIM

This is a graduate level course on advanced topics in probability and random variables. It aims to develop the ability to construct and analyze probabilistic models in a manner that combines intuitive understanding and mathematical precision. Different from the intrductory probability course, the course starts with digging the foundations in probability theory, random variables, expectation, and then it covers advanced topics such as transforms of distribution, further topics in random variables, limit theorems, statistical inference. This course also intends to provide students with the selected topics in stochastics processes such as Poisson process, Renewal process, Galton-Watson process, Gaussian process and discrete Markov Chains.

CONTENT

This course contains; Review of Basic Concepts (Probability Triple, Classical Probability Spaces, Concept of Sigma Field, Probability Measure, Conditional Probability, Limits of Events),Measurable Functions, Random Variables, Distribution Function,Random Vector, Joint Distribution, Independence,Expectation, Integral, and Weak and Strong Convergence,Transforms of Distribution (Characteristic Functions, Moment Generating Functions),Common Families of Distributions,Derived distributions,Midterm preparation overview,Covariance and correlation, conditional expectation and variance, sums of random variables,Limit theorems,Statistical inference,Topics in theory of stochastic processes,Discrete Markov Chains-1,Discrete Markov Chains-2.

1. 1

   Uses the foundations of probability theory and random variables in mathematical problems.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

2. 2

   Applies the concept of expectation, integral and convergence from different perspectives to engineering problems.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

3. 3

   Applies distributions of functions of random variables and their transforms into engineering problems.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

4. 4

   Obtain statistical inference from a given data set.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

5. 5

   It analyzes the performance of the system with Markov chains.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

6. 6

   Analyzes statistical images.

   Taught by: Simulation Technique, Lecture Method · Assessed by: Traditional Written Exam, Project Task

WEEKLY PLAN

1. WEEK 1

   Review of Basic Concepts (Probability Triple, Classical Probability Spaces, Concept of Sigma Field, Probability Measure, Conditional Probability, Limits of Events)

   Preparation: Lecture Notes, Chapter 1 of Textbook 1

2. WEEK 2

   Measurable Functions, Random Variables, Distribution Function

   Preparation: Chapter 1 of Textbook 1

3. WEEK 3

   Random Vector, Joint Distribution, Independence

   Preparation: Chapter 1 of Textbook 1

4. WEEK 4

   Expectation, Integral, and Weak and Strong Convergence

   Preparation: Chapter 2 of Textbook 1

5. WEEK 5

   Transforms of Distribution (Characteristic Functions, Moment Generating Functions)

   Preparation: Bölüm 3 Textbook 1

6. WEEK 6

   Common Families of Distributions

   Preparation: Chapter 3 of Textbook 1

7. WEEK 7

   Derived distributions

   Preparation: Chapter 4 of Textbook 2

8. WEEK 8

   Midterm preparation overview

   Preparation: All the topics till Week 8.

9. WEEK 9

   Covariance and correlation, conditional expectation and variance, sums of random variables

   Preparation: Chapter 4 of Textbook 2

10. WEEK 10

    Limit theorems

    Preparation: Chapter 5 of Textbook 2

11. WEEK 11

    Statistical inference

    Preparation: Chapter 9 of Textbook 2

12. WEEK 12

    Topics in theory of stochastic processes

    Preparation: Chapter 8 of Textbook 1, Chapter 6 of Textbook 2

13. WEEK 13

    Discrete Markov Chains-1

    Preparation: Chapter 8 of Textbook 1, Chapter 7 of Textbook 2

14. WEEK 14

    Discrete Markov Chains-2

    Preparation: Chapter 8 of Textbook 1, Chapter 7 of Textbook 2

ASSESSMENT

• Rate of Midterm Exam to Success50%
• Rate of Final Exam to Success50%

WORKLOAD

READING

• 1. Advanced Probability Theory (Probability: Pure and Applied) , Janos Galambos, ISBN-13:978-9052016580
• 2. Introduction to Probability, 2nd Ed., Dimitri P. Bertsekas and John N. Tsitsiklis, ISBN-13: 978-1886529236

TEACHING STAFF

• Prof.Dr. Mehmet Kemal ÖZDEMİRCOORDINATOR
• Lect.Dr. Ali Tuğberk DOĞUKAN