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 | Area Elective | 3 | 0 | 0 | 3 | 5 |
| Pre-requisite Course(s) |
|---|
| N/A |
| Course Language | English |
|---|---|
| Course Type | Elective Courses |
| Course Level | Natural & Applied Sciences Master's Degree |
| Mode of Delivery | Face To Face |
| Learning and Teaching Strategies | Lecture. |
| Course Lecturer(s) |
|
| 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;
|
| 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 | - |
Sources
| 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 | 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. | X | ||||
| 2 | Has the ability to obtain, to evaluate, to interpret and to apply information by doing scientific research. | X | ||||
| 3 | Can apply gained knowledge and problem solving abilities in inter-disciplinary research. | X | ||||
| 4 | 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. | X | ||||
| 5 | 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. | X | ||||
| 6 | Can develop strategies, implement plans and principles on the area of study and can evaluate obtained results within the framework. | X | ||||
| 7 | Can develop and extend the knowledge in the area and to use them with scientific, social and ethical responsibility. | X | ||||
| 8 | 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. | X | ||||
| 9 | Has proficiency in English language and has the ability to communicate with colleagues and to follow the innovations in mathematics and related fields. | X | ||||
| 10 | Has software and hardware knowledge in the area of expertise, and has proficient information and communication technology knowledge. | X | ||||
| 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. | X | ||||
ECTS/Workload Table
| Activities | Number | Duration (Hours) | Total Workload |
|---|---|---|---|
| Course Hours (Including Exam Week: 16 x Total Hours) | 16 | 3 | 48 |
| Laboratory | |||
| Application | |||
| Special Course Internship | |||
| Field Work | |||
| Study Hours Out of Class | 16 | 2 | 32 |
| Presentation/Seminar Prepration | |||
| Project | 2 | 12 | 24 |
| Report | |||
| 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 | ||