# Stochastic Processes (MATH495) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Stochastic Processes MATH495 Area Elective 3 0 0 3 6
Pre-requisite Course(s)
Math 392 or Consent of the instructor
Course Language English Elective Courses Bachelor’s Degree (First Cycle) Face To Face Lecture, Question and Answer, Problem Solving. Prof. Dr. Sofiya Ostrovska Asst. Prof. Dr. Ümit Aksoy This course is intended primarily for the student of mathematics, physics or engineering who wishes to learn the notion of stochastic processes and get familiar with their common applications. The students who succeeded in this course; At the end of the course the students are expected to: 1) Know the properties and usage of special probability distributions such as Erlang, Weibull, hypoexponential. 2) Understand the notion of stochastic process and analyze different types of stochastic processes. 3) know the Poisson process, its properties, applications and generalizations. 4) classify states and compute probabilities for Markov Chains 5) Model different real-life situations with the help of stochastic processes. Basic notions of probability theory; reliability theory; notion of a stochastic process; Poisson processes, Markov chains; Markov decision processes.

### Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Preliminaries: Probability, random events and random variables. Independence. pp. 1 - 10
2 Classical probability distributions, their properties. Random vectors. Coditional distribution and conditional expectation. pp. 11 -14
3 Reliability theory. Finding reliability function for different systems. Redundancy. [1], pp. 29-33,pp 124-135.
4 Hazard rate function, the mean time to failure. [1], pp. 228-236
5 Definition and examples of stochastic processes, their types. pp. 26-27, [1], pp. 294-300
6 The Bernoulli and Poisson processes. Interarrival and waiting times. pp. 31-36
7 Non-homogeneous and compound Poisson processes. Midterm I pp. 46 - 49
8 Renewal processes. Erlang process. Renewal theorems. pp. 55-60
9 Markov chains: Markov property, transition probabilities, transition graph. The Chapman-Kolmogorov equations.Computation of n-th step transition probabilities. pp. 100-103
10 Classification of states and limiting probabilities. Equlibrium. pp. 104-110
11 Absorbing Markov chains. Fundamental matrix. [1], pp. 392-402
12 Midterm II. Continuous-time Markov chains. Kolmogorov’s equations. pp.141-150
13 Time reversibility. pp. 156-158
14 Applications of Markov chains. pp. 118-122
15 Review.
16 Final exam.

### Sources

Course Book 1. Sheldon M. Ross, Stochastic processes, Wiley, 1983. 2. K. S. Trivedi, Probability and Statistics with Reliability, Queueing, and Computer Science Applications, 2nd Edition, Wiley, 2002. 3. J. G. Kemeny and J. L. Snell, Finite Markov chains, Springer, 1976. 4. S. Karlin, H. M. Taylor, A first course in stochastic processes, 2-nd Ed, Academic Press, 1975.

### Evaluation System

Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments 4 20
Presentation - -
Project - -
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 40
Final Exam/Final Jury 1 40
Toplam 7 100
Percentage of Semester Work 100 100

### Course Category

Core Courses X

### The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Has the ability to apply scientific knowledge gained in the undergraduate education and to expand and extend knowledge in the same or in a different area
2 Can apply gained knowledge and problem solving abilities in inter-disciplinary research
3 Has the ability to work independently within research area, to state the problem, to develop solution techniques, to solve the problem, to evaluate the obtained results and to apply them when necessary
4 Takes responsibility individually and as a team member to improve systematic approaches to produce solutions in unexpected complicated situations related to the area of study
5 Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework
6 Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility
7 Has the ability to follow recent developments within the area of research, to support research with scientific arguments and data, to communicate the information on the area of expertise in a systematically by means of written report and oral/visual presentation
8 To have an oral and written communication ability in at least one of the common foreign languages ("European Language Portfolio Global Scale", Level B2)
9 Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge
10 Follows scientific, cultural, and ethical criteria in collecting, interpreting and announcing data in the research area and has the ability to teach.
11 Has professional ethical consciousness and responsibility which takes into account the universal and social dimensions in the process of data collection, interpretation, implementation and declaration of results in mathematics and its applications.

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours)
Laboratory
Application
Special Course Internship
Field Work
Study Hours Out of Class 16 3 48
Presentation/Seminar Prepration
Project
Report
Homework Assignments 4 10 40
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 12 24
Prepration of Final Exams/Final Jury 1 18 18