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.
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.