ECTS - Analytical Probability Theory

Analytical Probability Theory (MDES615) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Analytical Probability Theory MDES615 3 0 0 3 5
Pre-requisite Course(s)
Consent of the instructor
Course Language English
Course Type N/A
Course Level Natural & Applied Sciences Master's Degree
Mode of Delivery Face To Face
Learning and Teaching Strategies Lecture.
Course Coordinator
Course Lecturer(s)
Course Assistants
Course Objectives The objective of the course is to study the properties of probability distributions and their applications with the help of analytic methods. The course is based on the modern approach to Probability Theory. Since engineering scientists need powerful analytic tools of Probability Theory to analyze algorithms and computer systems, a great number of practical examples are included into the course.
Course Learning Outcomes The students who succeeded in this course;
  • Understand basic notions of Probability Theory Model real-life situations with random outcomes and have knowledge of classical probability distributions. Know fundamentals of reliability theory and simulation of probability distributions. Analyze different types of probability distributions and decompose mixed distributions. Apply the transform methods to finding distributions for sums of independent random variables and limit distributions.
Course Content Sigma-algebra of sets, measure, integral with respect to measure; probability space; independent events and independent experiments; random variables and probability distributions; moments and numerical characteristics; random vectors and independent random variables; convergence of random variables; transform methods; sums of independent random v

Weekly Subjects and Releated Preparation Studies

Week Subjects Preparation
1 Sigma-algebra of sets, measure, measurable functions. Integral with respect to measure Ch.1.1-1.7
2 Probability space. Basic properties of probability. Independent and dependent events. Pairwise independence, independence at level k, stochastic independence. Ch. 1.9-1.11
3 Introduction to the reliability theory: reliability of series-parallel systems and non-series-parallel systems. Independent experiments. Bernoulli trials. Reliability of an m-out-of-n system. Ch. 1.12
4 Random variables, their distributions. Distribution function. The probability mass function and probability density. Ch. 2.1, 2.2, 2.4
5 Pure and mixed type distributions. Lebesgue decomposition theorem. Ch. 3.1
6 Classical probability distributions, their properties and applications. The usage of Poisson distribution. Ch. 2.5, 3.4
7 Memoryless property of the exponential distribution. Reliability function. Ch. 3.2, 3.3
8 Functions of random variables, their distributions. Numerical characteristics of random variables. Moments. Chebyshev inequality. Ch. 4.1, 4.2
9 Random vectors. Distribution of a random vector and distribution of components Ch. 2.9, 3.6
10 Independent random variables, their properties. Conditional distribution and conditional expectation. Ch. 5.1, 5.2, 5.3
11 Independent random variables, their properties. The convolution theorem. Erlang distribution. Ch. 2.9
12 Transform methods: Moment generating functions, their properties and applications. Ch. 4.5
13 Sums of independent random variables. Hypoexponential distribution. Standby redundancy. Ch. 3.8
14 Convergence in distribution. Limit distribution. The central limit theorem Ch. 4.7
15 Overall review -
16 Final exam -


Course Book 1. K. S. Trivedi, Probability and Statistics with Reliability, Queueing, and Computer Science Applications, 2nd Edition, Wiley, 2002.
Other Sources 2. W.Feller. An Introduction to probability theory and its applications, v.I,II. J.Wiley and Sons, New-York, 1986
3. K.L. Chung. A Course in Probability Theory Revised. Acad. Press, 3rd Ed.
4. M.H. DeGroot, M.J. Shervish. Probability and Statistics. Addison Wesley, 2002

Evaluation System

Requirements Number Percentage of Grade
Attendance/Participation - -
Laboratory - -
Application - -
Field Work - -
Special Course Internship - -
Quizzes/Studio Critics - -
Homework Assignments - -
Presentation - -
Project 2 20
Report - -
Seminar - -
Midterms Exams/Midterms Jury 2 40
Final Exam/Final Jury 1 40
Toplam 5 100
Percentage of Semester Work 60
Percentage of Final Work 40
Total 100

Course Category

Core Courses X
Major Area Courses
Supportive Courses
Media and Managment Skills Courses
Transferable Skill Courses

The Relation Between Course Learning Competencies and Program Qualifications

# Program Qualifications / Competencies Level of Contribution
1 2 3 4 5
1 Ability to expand and get in-depth information with scientific researches in the field of mechanical engineering, evaluate information, review and implement.
2 Have comprehensive knowledge about current techniques and methods and their limitations in Mechanical engineering.
3 To complete and apply knowledge by using scientific methods using uncertain, limited or incomplete data; use information from different disciplines.
4 Being aware of the new and developing practices of Mechanical Engineering and being able to examine and learn when needed.
5 Ability to define and formulate problems related to Mechanical Engineering and develop methods for solving and apply innovative methods in solutions.
6 Ability to develop new and/or original ideas and methods; design complex systems or processes and develop innovative/alternative solutions in the designs.
7 Ability to design and apply theoretical, experimental and modeling based researches; analyze and solve complex problems encountered in this process.
8 Work effectively in disciplinary and multi-disciplinary teams, lead leadership in such teams and develop solution approaches in complex situations; work independently and take responsibility.
9 To establish oral and written communication by using a foreign language at least at the level of European Language Portfolio B2 General Level.
10 Ability to convey the process and results of their studies systematically and clearly in written and oral form in national and international environments.
11 To know the social, environmental, health, security, law dimensions, project management and business life applications of engineering applications and to be aware of the constraints of their engineering applications.
12 Ability to observe social, scientific and ethical values in the stages of data collection, interpretation and announcement and in all professional activities.

ECTS/Workload Table

Activities Number Duration (Hours) Total Workload
Course Hours (Including Exam Week: 16 x Total Hours) 16 3 48
Special Course Internship
Field Work
Study Hours Out of Class 16 2 32
Presentation/Seminar Prepration
Project 2 12 24
Homework Assignments
Quizzes/Studio Critics
Prepration of Midterm Exams/Midterm Jury 2 8 16
Prepration of Final Exams/Final Jury 1 10 10
Total Workload 130