TU Delft
Education Type
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2010/2011 Electrical Engineering, Mathematics and Computer Science Bachelor Electrical Engineering
Stochastic Processes
Responsible Instructor
Name E-mail
Dr. D.M.J. Tax    D.M.J.Tax@tudelft.nl
Name E-mail
Z. Erkin    Z.Erkin@tudelft.nl
Prof.dr.ir. R.L. Lagendijk    R.L.Lagendijk@tudelft.nl
Name E-mail
Dipl.ing. S. Rudinac    S.Rudinac@tudelft.nl
Contact Hours / Week x/x/x/x
0/0/2/0 lectures; 0/0/4/0 working groups; 0/0/4/0 lab work
Education Period
Start Education
Exam Period
Course Language
Expected prior knowledge
This course builds heavily on Mathematics (especially integration and differentiation), Signal Processing and Transformations (especially linear systems and signals, Fourier analysis).
Course Contents
In this course we will introduce the concept of stochastic models and random processes for describing systems and signals that are not deterministic. In fact, no single system or signal is deterministic in practice. For that reason the theory of stochastics and random processes should be considered as a useful extension of known approaches for describing and modeling systems and signals encountered in engineering practices.

Stochastic models will be developed on the basis of probability theory. Probability theory describes the behavior of certain phenomena in terms of how likely it is that certain values will occur. Central features of the models will be discussed are random variables, probability density functions, and the expected value operator. In describing random processes and signals, the correlation function and conditional probabilities play a central role.

The course addresses the following subjects:
1. Refresher random variables. Matlab exercise on estimation of PDF, expected value and variance.
2. Refresher correlation. Calculating with correlation functions.
3. Random processes, correlation function, stationarity, wide sense stationarity, estimation of correlation function (Matlab exercise).
4. Random signal processing, power spectral density function, white noise.
5. AR processes, linear prediction: theory and Matlab exercise.
6. Markov chains.
7. Matlab exercise based on applications of Markov chain.
Study Goals
1. Probability Theory
- Conditional) probabilities, the law of total probability, and Bayes’ rule.
- Solve probability problems that require the use of axioms of probability.

2. Definition and Description of Random Variables and Processes
PDF, PMF, CDF, Covariance, Correlation- Determine if a given PDF, PMF, CDF, variance, (auto/cross-)correlation(-function), (auto/cross-)covariance(-function), power spectral density complies with (theoretical and analytical) requirements.
- Convert the description of a probabilistic problem into a probabilistic model using PDF, PMF, or CDF.

3. PDF/PMF and Expected Value
Calculate the various forms of expected value of (combinations of) random variables and random processes
- For a given (amplitude continuous/discrete and time continuous/discrete) probability model calculate the following probabilistic (marginal, joint and conditional) characterizations: PDF, PMF, CDF, probability of an event, expected value, variance, covariance, correlation, correlation coefficient, auto/crosscorrelation function, auto/crosscovariance function, (cross) power spectral density.
- Calculate the PDF, PMF, expected value and variance of a derived random variable.

4. Properties of Random Processes
- Independence, orthogonality, uncorrelated, whiteness, IID- Determine if random variables/processes have the following properties: independent, orthogonal, uncorrelated, white, Poisson, Gaussian, Bernoulli, Markov, IID, stationary, WSS, ergodic.
- Calculate the expected value, variance, auto/crosscorrelation(function), auto/crosscovariance(function), power spectral density of a linear combination of random variables and of a linearly filtered (WSS, amplitude discrete/continuous, time discrete/continuous) random process.

5. Large NumbersCentral limit theorem, law of large numbers
- Solve problems that require the use of the central limit theorem in an engineering context
- Explain the law of the large numbers in an engineering context.

6. Statistical Estimators
- Estimated mean, variance, and correlation function
- Given a set of outcomes, sample functions or realizations, calculate estimators for expected value, variance, and (auto-)correlation function.

7. Application to Engineering Problems and Simulations
- Give examples of problems in simple electrical engineering or computer science (media and knowledge engineering) where probability theory and random processes is useful.
- Select and translate a simple electrical engineering or computer science problem into mathematical probability model. The emphasis is on problems in signal and image processing, telecommunication, and media and knowledge technology. The class of probability models encompasses the following random variables/processes: Bernoulli, exponential, binomial, Poisson, Gaussian, uniform.
- Justify and reflect on the approach taken in calculating or simulating (MatLab) the following probabilistic properties: PDF, PMF, expected value, variance, autocorrelation function, autocovariance function.
Education Method
Lectures, working groups (problem solving), laboratory work (Matlab exercises)
Literature and Study Materials
R.D. Yates and D.J. Goodman, "Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers", ISBN 0-471-17837-3, John Wiley and Sons, New York, 2005, Second Edition.
R.D. Yates and D.J. Goodman, "Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers", ISBN 0-471-17837-3, John Wiley and Sons, New York, 2005, Second Edition.
(1) Written (open question)
(2) Oral exam (10 minutes) about 7-th Matlab exercises (with pass/fail result)
Permitted Materials during Tests
It is allowed to use the course textbook and the Overview Table of Definitions during the written exam.
To pass the exam, two requirements need to be fulfilled.

(1) Solve the Matlab Exercise session 7. The answers to this problem must be presented to the lecturers in a short oral exam session in week 8. The result of this oral exam is a pass or fail. Only if a pass is obtained, the result of the written exam Stochastic Processes will become valid.

(2) Pass the written exam. Exam question are similar to working group and recommended questions in the text book, the "sample exam" questions that are solved during the working group, and the Matlab exercise questions.