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CSCI 635  :   Probability and Stochastic Processes

Course  Name:  CSCI  635

Probability and Stochastic Processes

Course Credits:  3

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 real-life 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.

Teaching method:
The presentation is problem-oriented and motivated by practical examples. Assignments are given which should be worked out independently and presented in the lecture.


Background : Pre-requisites:  Good background in  mathematics  and Knowledge of Undergraduate Probability Theory .

Pre-requisite : 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.

Other Recommended Book  (Reference Texts ):

  • Signals and Systems -A.V.Oppenheim,A.S.WillskyandI.T.Young.

  • Probability and measure theory (Second Edn.), R.B. Ash and C.A. Doleans-Dade. Harcourt Academic Press, ISBN 0-12-065202-1.

  • A first course in stochastic processes (Second Edn.), S. Karlin and H.M. Taylor,Academic Press, ISBN 0-12-398552-8.

  • Probability, random variables, and stochastic processes, A. Papoulis, New York : McGraw-Hill.

  • Probability and random processes with applications to signal processing, 3rd Edn., H. Stark and J.W. Woods, Prentice Hall, ISBN 0-13-020071-9.

  • Probability, A.N. Shiryaev (Second Edn.), Springer, ISBN 0-387-94549-0

  • Fun and Games, K. Binmore, D.C. Heath and Company, ISBN 0-669-24603-4


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).


Topics to be Covered: 

The  CSCI 635   will cover the following topics :


  1. Introduction to the Course.


  1. Elements of Probability Theory.

    • Deterministic and random.

    • Random variables.

    • Noise as stochastic processes


  1. 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.

  2. Application To Electrical Engineering-noise Analysis.

    • Applications to performance analysis of CW communication systems.

    • Introduction to queuing and information theory.

    • Applications to digital communication system.

Marking Scheme:

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

  • Exams are open book.

  • There will be no make-up exams.

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

  • Negative marks will be given to copiers and copies.

Pre-information :

  • Introduction to noises.


Mark distribution :


Mid-Term  Test                                       30%

Assignments                                           30%

Final                                                        40%

Assignments :


  • Probability and Random Variables.

  • Linear Estimation Theories.

  • Communication Channels Example.

  • Recursive Least-squares Algorithm( Kalman filter ).

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) DeMoivre-Laplace Theorem

14) Law of Large Numbers

15) Poisson Theorem and Random Points.

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.

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.

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 Multi-dimensional Gaussian Density.

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 Second-Order Probabilistic Averages

6) Wide-Sense 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.

Part 7: Correlation Functions

1) Autocorrelation Function

2) Auto-covariance Function

3) Correlation Function

4) Properties of Autocorrelation Function for Real-Valued WSS Random Processes

5) Random Binary Waveform

6) Poisson Random Points (Revisited)

7) Poisson Random Processes

8) Autocorrelation of Poisson Processes

9) Semi-Random Telegraph Signal

10) Random Telegraph Signal

11) Autocorrelation of Wiener Process

12) Correlation Time

13) Cross-correlation 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) Wiener-Khinchine 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 Low-pass 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:

Attribute: Contributions from this Course: In-depth 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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