# 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 N/A Natural & Applied Sciences Master's Degree Face To Face Lecture. 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. 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. 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. 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 40 100

### 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 Accumulated knowledge on mathematics, science and mechatronics engineering; an ability to apply the theoretical and applied knowledge of mathematics, science and mechatronics engineering to model and analyze mechatronics engineering problems.
2 An ability to differentiate, identify, formulate, and solve complex engineering problems; an ability to select and implement proper analysis, modeling and implementation techniques for the identified engineering problems.
3 An ability to design a complex system, product, component or process to meet the requirements under realistic constraints and conditions; an ability to apply contemporary design methodologies; an ability to implement effective engineering creativity techniques in mechatronics engineering. (Realistic constraints and conditions may include economics, environment, sustainability, producibility, ethics, human health, social and political problems.)
4 An ability to develop, select and use modern techniques, skills and tools for application of mechatronics engineering and robot technologies; an ability to use information and communications technologies effectively.
5 An ability to design experiments, perform experiments, collect and analyze data and assess the results for investigated problems on mechatronics engineering and robot technologies.
6 An ability to work effectively on single disciplinary and multi-disciplinary teams; an ability for individual work; ability to communicate and collaborate/cooperate effectively with other disciplines and scientific/engineering domains or working areas, ability to work with other disciplines.
7 An ability to express creative and original concepts and ideas effectively in Turkish and English language, oral and written.
8 An ability to reach information on different subjects required by the wide spectrum of applications of mechatronics engineering, criticize, assess and improve the knowledge-base; consciousness on the necessity of improvement and sustainability as a result of life-long learning; monitoring the developments on science and technology; awareness on entrepreneurship, innovative and sustainable development and ability for continuous renovation.
9 Be conscious on professional and ethical responsibility, competency on improving professional consciousness and contributing to the improvement of profession itself.
10 A knowledge on the applications at business life such as project management, risk management and change management and competency on planning, managing and leadership activities on the development of capabilities of workers who are under his/her responsibility working around a project.
11 Knowledge about the global, societal and individual effects of mechatronics engineering applications on the human health, environment and security and cultural values and problems of the era; consciousness on these issues; awareness of legal results of engineering solutions.
12 Competency on defining, analyzing and surveying databases and other sources, proposing solutions based on research work and scientific results and communicate and publish numerical and conceptual solutions.
13 Consciousness on the environment and social responsibility, competencies on observation, improvement and modify and implementation of projects for the society and social relations and be an individual within the society in such a way that planing, improving or changing the norms with a criticism.
14 A competency on developing strategy, policy and application plans on the mechatronics engineering and evaluating the results in the context of qualitative processes.

### 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