Each module contains 3 ECTS. You choose a total of 10 modules/30 ECTS in the following module categories:

- 12-15 ECTS in technical scientific modules (TSM)

TSM modules teach profile-specific specialist skills and supplement the decentralised specialisation modules. - 9-12 ECTS in fundamental theoretical principles modules (FTP)

FTP modules deal with theoretical fundamentals such as higher mathematics, physics, information theory, chemistry, etc. They will teach more detailed, abstract scientific knowledge and help you to bridge the gap between abstraction and application that is so important for innovation. - 6-9 ECTS in context modules (CM)

CM modules will impart additional skills in areas such as technology management, business administration, communication, project management, patent law, contract law, etc.

In the module description (download pdf) you find the entire language information per module divided into the following categories:

- instruction
- documentation
- examination

The ubiquitous presence of uncertainty and noise in the engineering sciences and the importance of randomized algorithms in computer and data science make it mandatory to understand and quantify random phenomena. To achieve this goal the course will provide a solid review of probability theory and an introduction to the theory of stochastic processes. Special attention is given to applications, including examples from various fields such as communications and vision, signal processing and control, queuing theory or physics of small systems (Brownian motion).

### Prerequisites

- Basis calculus (integration, differentiation, ordinary differential equations, complex numbers, Fourier transform)
- Basic probability theory (probability, conditional probability, Bayes' theorem, expectation, variance, random variables)
- Linear algebra (matrix algebra, system of linear equations, eigenvectors, eigenvalues)

### Learning Objectives

The student is familiar with the main working tools and concepts of stochastic modeling (expectation, variance, covariance, autocorrelation, power spectral density). He/She is able to explain properties and limitations of stochastic processes as a modeling tool for noisy systems. He/She will be able to model and analyze simple random phenomena through adaptation of proposed stochastic models.

### Contents of Module

- Probability review: random variables, conditional probabilities, theorem of large numbers, central limit theorem.
- General introduction to discrete and continuous stochastic processes. Applications, e.g., communications, Kalman filtering.
- Discrete, continuous and hidden Markov chains. Applications, e.g., page rank algorithm, queuing systems, pattern recognition, speech recognition.
- Bernoulli, Poisson, Gaussian processes, Brownian motion, white and colored noise.

### Teaching and Learning Methods

Ex cathedra teaching

Presentation of simulation results and case studies

### Literature

The script is, in principle, sufficient. Further readings are:

- Sheldon M. Ross, Probability Models, Elsevier.
- John A. Gubner, Probability and Random Processes for Electrical and Computer Engineers, Cambridge University Press.
- Mario Lefebvre, Applied Stochastic Processes, Springer.
- Bassel Solaiman, Processus stochastiques pour l’ingénieur, PPUR.

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