MSc in Electrical Engineering

## Course Information Package

Course Unit TitleRANDOM VARIABLES AND STOCHASTIC PROCESSES
Course Unit CodeAEEE503
Course Unit DetailsMSc Electrical Engineering (Required Courses) -
Number of ECTS credits allocated7
Learning Outcomes of the course unitBy the end of the course, the students should be able to:
1. Present Bayes’ theorem and examine applications related to communication over noisy channels.
2. Define Random Variables and related concepts such as the Probability Distribution Function, the Probability Density Function, the Expected Value of a Random Variable, Conditional Expectations, Moments, Moment Generating Functions and Characteristic Functions.
3. Appraise the importance of the Gaussian Random Variable and introduce the univariate Normal (Guassian) probability density function.
4. Analyze Functions of Random Variables with reference to concepts such as conditional and joint distributions and densities.
5. Review key concepts of Linear Algebra: Multiplication, Linear Dependence, Determinants, Eigenvalues, Eigenvectors, Positive Definite Matrices, Causal Factorization, Spectral Resolution.
6. Define the Covariance and Correlation, analyze Linear Transformation of Random Vectors and formulate the Simulation Problem.
7. Introduce Gaussian Functions, Gaussian Characteristic Functions, Linear Transformations of Gaussian functions and the probability density function of a Gaussian random vector.
8. Perform Hypothesis Testing with second order information and investigate Correlation Detection in Additive Noise and Whitening,
9. Present Bayes decision theory and explore applications such as minimization of probability of error, Likelihood ratio tests and Mean Square Estimation.
10. Define Random Processes with reference to specific examples.
11. Analyze applications and concepts related to random processes such as Phase Shift Keying, Wiener Processes, Markov Processes, Poisson Processes, Stationarity, Power Spectral Density and Kalman Filtering.
Mode of DeliveryFace-to-face
PrerequisitesNONECo-requisitesNONE
Recommended optional program componentsNONE
Course Contents

Introduction to Probability : Overview of set theory. Sample Spaces, Events, Sigma-fields, Axiomatic Definition of Probability, Joint Probabilities, Conditional Probabilities, Total Probability, Independence, Bayes’ theorem and applications, Communication over noisy channels.

Random Variables: Definition of Random Variables, Probability distribution function, Probability density function, Conditional and joint distributions and densities, Functions of Random Variables, Expected Value of a Random Variable, Conditional Expectations, Moments, Joint Moments, Moment Generating Functions, Characteristic Functions.

Revision on Linear Algebra: Multiplication, Linear Dependence, Determinants, Eigenvalues, Eigenvectors, Positive Definite Matrices, Causal Factorization, Spectral Resolution.

Second Moment Descriptions: Covariance, Correlation, Linear Transformation of Random Vectors, the Simulation Problem, Gaussian Functions, Gaussian Characteristic Functions, Linear Transformations, The probability density function of a Gaussian random vector.

Applications using Second Order Information: Hypothesis Testing with second order information, Correlation Detection in Additive Noise, Whitening, Bayes decision theory, Minimization of probability of error, Likelihood ratio tests, Mean Square Estimation.

Stochastic Processes: Definition of Random Processes, Examples of Random Processes, Phase Shift Keying, Wiener Process, Markov Processes, Poisson Processes, Stationarity, Power Spectral Density, Kalman Filtering.

Textbooks
• Henry Stark, John W. Woods, Probability and Random Processes, Prentice Hall, 2002.
References
• Alberto Leon-Garcia, Probability and Random Processes for Electrical Engineering, Addison Weseey, 1994.
• Athanasios Papoulis, S. Unnikrishna Pillai, Probability, Random Variables and Stochastic Processes, McGraw-Hill Science, 2001.
Planned learning activities and teaching methods

Students are taught the course through lectures (3 hours per week) in classrooms or lectures theatres, by means of traditional tools or using computer demonstration.

Auditory exercises, where examples regarding matter represented at the lectures, are solved and further, questions related to particular open-ended topic issues are compiled by the students and answered, during the lecture or assigned as homework.

Topic notes are compiled by students, during the lecture which serve to cover the main issues under consideration and can also be downloaded from the lecturer’s webpage. Students are also advised to use the subject’s textbook or reference books for further reading and practice in solving related exercises. Tutorial problems are also submitted as homework and these are solved during lectures or privately during lecturer’s office hours. Further literature search is encouraged by assigning students to identify a specific problem related to some issue, gather relevant scientific information about how others have addressed the problem and report this information in written or orally.

Students are assessed continuously and their knowledge is checked through tests with their assessment weight, date and time being set at the beginning of the semester via the course outline.

Students are prepared for final exam, by revision on the matter taught, problem solving and concept testing and are also trained to be able to deal with time constraints and revision timetable.

The final assessment of the students is formative and summative and is assured to comply with the subject’s expected learning outcomes and the quality of the course.

Assessment methods and criteria
 Assignments 10% Tests 30% Final Exam 60%
Language of instructionEnglish
Work placement(s)NO