Prof. Dr. Ebrahim A. Mattar
Professor of Robotics/Cybernetics
College of Engineering, University of Bahrain
Research Interests:
Robotics, Cybernetics, AI
Now working on Electroencephalography (EEG) Brainwaves Decoding for Building Robotics Cognition
CSCI 635 : Probability and Stochastic Processes
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Course Name: CSCI 635
Probability and Stochastic Processes
Course Credits: 3
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Course Aims and objectives:
To provide a solid introduction to probability theory and stochastic processes. With this background the students should be able to specify and solve simple probabilistic problems. They should gain also some practice in working with basic stochastic models. Knowledge of stochastic processes is essential for the study/analysis of computer networks, wireless communications, multimedia systems, financial market etc. This course is an introduction to the theory of stochastic (random) processes and its applications to many reallife problems. No prior knowledge of stochastic processes will be assumed. At the end of the course students will be able to analyze communication and networking systems, predict financial markets etc. Fun part of this course will include the study of some bizarre problems in stochastic processes.
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Teaching method:
The presentation is problemoriented and motivated by practical examples. Assignments are given which should be worked out independently and presented in the lecture.
Background : Prerequisites: Good background in mathematics and Knowledge of Undergraduate Probability Theory .
Prerequisite : Probability and Stochastic Processes or equivalent. Knowledge of random variables, functions of random variables, conditional probability, and expected value will be assumed.
Remarks :
Homework must be submitted individually (but, discussions with other students in the class are allowed). Late submissions will be penalized. Feel free to ask questions in the class. The course website will be updated periodically for homework problems, announcements about assignments and exams etc.!!
Textbook :
Students are not expected to purchase the following books, but may find them useful. Some are textbooks from previous courses, and copies of these books are available in the library. Other texts on this area, using Java and other languages, are available in some library.
Probability and Stochastic Processes : A Friendly Introduction for Electrical and Computer Engineers
Roy D. Yates and David J. Goodman, Publisher: John Wiley & Sons, Matlab Demos.
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Other Recommended Book (Reference Texts ):
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Signals and Systems A.V.Oppenheim,A.S.WillskyandI.T.Young.

Probability and measure theory (Second Edn.), R.B. Ash and C.A. DoleansDade. Harcourt Academic Press, ISBN 0120652021.

A first course in stochastic processes (Second Edn.), S. Karlin and H.M. Taylor,Academic Press, ISBN 0123985528.

Probability, random variables, and stochastic processes, A. Papoulis, New York : McGrawHill.

Probability and random processes with applications to signal processing, 3rd Edn., H. Stark and J.W. Woods, Prentice Hall, ISBN 0130200719.

Probability, A.N. Shiryaev (Second Edn.), Springer, ISBN 0387945490

Fun and Games, K. Binmore, D.C. Heath and Company, ISBN 0669246034
Course Software: The software which is needed for that course is : MATLAB. You can use C++ programming if you need to.
Distribution of Notices and Course Handouts
Handouts to each student will be made available soon. We will mainly use my lecture notes and book chapters (as you will get them later).
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Topics to be Covered:
The CSCI 635 will cover the following topics :

Introduction to the Course.

Elements of Probability Theory.

Deterministic and random.

Random variables.

Noise as stochastic processes


Random Variables

Functions of Random Variables and Limit Theorems.

Spectral Density.

Ergodicity and applications in Linear Systems.

Spectral analysis and estimation.

Estate estimation and filtering (Kalman filter,Wiener filter .....).

Wiener Processes.

Mean Square Estimation and Markov Process.



Application To Electrical Engineeringnoise Analysis.

Applications to performance analysis of CW communication systems.

Introduction to queuing and information theory.

Applications to digital communication system.

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Marking Scheme:

There will be , a midterm exam worth , and a final exam worth .

Exams are open book.

There will be no makeup exams.

Discussions among students are encouraged but all assignments must be done independently.

Negative marks will be given to copiers and copies.
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Preinformation :

Introduction to noises.
Mark distribution :
MidTerm Test 30%
Assignments 30%
Final 40%
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Assignments :

Probability and Random Variables.

Linear Estimation Theories.

Communication Channels Example.

Recursive Leastsquares Algorithm( Kalman filter ).
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Part 1: Introduction to Probability
1) The Classical Approach to Probability
2) The Relative Frequency Approach to Probability
3) The Axiomatic Approach to Probability
4) Elementary Set Theory
5) Probability Space: Sample Space, ? Algebra and Probability Measure
6) Conditional Probability
7) Theorem of Total Probability  Discrete Form
8) Bayes Theorem
9) Independence of Events
10) Cartesian Product of Sets
11) Independent Bernoulli Trials
12) Gaussian Function
13) DeMoivreLaplace Theorem
14) Law of Large Numbers
15) Poisson Theorem and Random Points.
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Part 2: Random Variables
1) Random Variables
2) Distribution and Density Function
3) Continuous/Discrete/Mixed Random Variables
4) Normal/Gaussian Random Variable
5) Uniform Random Variable
6) Binomial Random Variable
7) Poisson Random Variable
8) Rayleigh Random Variable
9) Exponential Random Variable
10) Conditional Distribution/Density
11) Theorem of Total Probability  Continuous Form
12) Bayes Theorem  Continuous Form
13) Expectation
14) Variance and Standard Deviation
15) Moments
16) Conditional Expectation
17) Tchebycheff Inequality
18) Poisson Points Applied to System Reliability.
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Part 3: Multiple Random Variables
1) Joint Distribution/Density
2) Jointly Gaussian Random Variables
3) Independence of Random Variables
4) Expectation of a Product of Random Variables
5) Variance of a Sum of Independent Random Variables
6) Random Vectors and Covariance Matrices.
Part 4: Function of Random Variables
1) Transformation of One Random Variable Into Another
2) Determination of Distribution/Density of Transformed Random Variables
3) Expected Value of Transform Random Variable
4) Characteristic Functions and Applications
5) Characteristic Function for Gaussian Random Vectors
6) Moment Generating Function
7) One Function of Two Random Variables
8) Leibnitz’s Rule
9) Two Functions of Two Random Variables
10) Joint Density Functions
11) Linear Transformation of Gaussian Random Variables.
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Part 5: Moments and Conditional Statistics
1) Expected Value of a Function of Two Random Variables
2) Covariance
3) Correlation Coefficient
4) Uncorrelated and Orthogonal Random Variables
5) Joint Moments
6) Conditional Distribution/Density: One Random Variable Conditioned on Another
7) Conditional Expectation
8) Application of Conditional Expectation: Bayesian Estimation
9) Conditional Multidimensional Gaussian Density.
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Part 6: Random Processes
1) Definitions and Examples of Random Processes
2) Continuous and Discrete Random Processes
3) Distribution and Density Functions
4) Stationary Random Processes
5) First and SecondOrder Probabilistic Averages
6) WideSense Stationary Processes
7) Ergodic Processes
8) Classical Random Walk
9) Wiener Process As a Limit of the Random Walk
10) Independent Increments
11) Diffusion Equation for Transition Density
12) Probability Current
13) Solution of Diffusion Equation by Transform Techniques.
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Part 7: Correlation Functions
1) Autocorrelation Function
2) Autocovariance Function
3) Correlation Function
4) Properties of Autocorrelation Function for RealValued WSS Random Processes
5) Random Binary Waveform
6) Poisson Random Points (Revisited)
7) Poisson Random Processes
8) Autocorrelation of Poisson Processes
9) SemiRandom Telegraph Signal
10) Random Telegraph Signal
11) Autocorrelation of Wiener Process
12) Correlation Time
13) Crosscorrelation Function
14) Input/Output Cross Correlation for Linear Systems
15) Autocorrelation of System Output in Terms of Autocorrelation of Input.
Part 8: Power Density Spectrum
1) Definition of Power spectrum of a Stationary Process
2) Calculation of Power spectrum of a Process
3) Rational Power Spectrums
4) WienerKhinchine theorem
5) Application to Random Telegraph Signal
6) Power Spectrum of System Output in Terms of Power Spectrum of System Input
7) Noise Equivalent Bandwidth of a Lowpass System or Filter.
Teaching Activities:
Classes: There is one class each week at 6 pm on Sunday, consisting of a formal component, followed by a demonstration related to the practical if any.
Tutorials: There are a number of tutorials as will be posted later.
Graduate Attributes Developed:
The University has defined a set of graduate attributes to specify broad core knowledge and skills associated with all undergraduate programs. This course addresses these attributes as follows:
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Attribute: Contributions from this Course: Indepth knowledge of the field of study Probabilities and Random Processes, Power Density Spectrum … Effective Communication, Working as a team on projects and assignments Independence and Creativity Generating ideas for and receiving feedback on project and assignments Critical Judgment. Defining and analyzing problems in project and assignments Ethical and Social Understanding Project and project mentoring provides philosophical and social context of discipline
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