# Analytical Probability Theory (MDES615) Course Detail

Course Name Course Code Season Lecture Hours Application Hours Lab Hours Credit ECTS
Analytical Probability Theory MDES615 Elective Courses 3 0 0 3 5
Pre-requisite Course(s)
Consent of the instructor
Course Language English Elective Courses Taken From Other Departments Ph.D. 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

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 Gains the ability to understand and apply knowledge in the fields of mathematics, science and basic sciences at the level of expertise.
2 Gains the ability to access wide and deep knowledge in the field of Engineering by doing scientific research with current techniques and methods, evaluate, interpret and implement the gained knowledge.
3 Being aware of the latest developments his/her field of study, defines problems, formulates and develops new and/or original ideas and methods in solutions.
4 Designs and applies theoretical, experimental, and model-based research, analyzes and interprets the results obtained at the level of expertise.
5 Gains the ability to use the applications, techniques, modern tools and equipment in his/her field of study at the level of expertise.
6 Designs, executes and finalizes an original work process independently.
7 Can work in interdisciplinary and interdisciplinary teams, lead teams, use the information of different disciplines together and develop solution approaches.
8 Pays regard to scientific, social and ethical values in all professional activities and acquires responsibility consciousness at the level of expertise.
9 Contributes to the literature by communicating the processes and results of his/her academic studies in written form or orally in national and international academic environments, communicates effectively with communities and scientific staff working in the field of specialization.
10 Gains the skill of lifelong learning at the level of expertise.
11 Communicates verbally and in written form using a foreign language at least at the European Language Portfolio B2 General Level.
12 Recognizes the social, environmental, health, safety, legal aspects of engineering applications, as well as project management and business life practices, being aware of the limitations they place on engineering applications.

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