Course syllabus adopted 2021-02-26 by Head of Programme (or corresponding).
Overview
- Swedish nameGrundläggande stokastiska processer
- CodeMVE170
- Credits7.5 Credits
- OwnerMPENM
- Education cycleSecond-cycle
- Main field of studyMathematics
- DepartmentMATHEMATICAL SCIENCES
- GradingTH - Pass with distinction (5), Pass with credit (4), Pass (3), Fail
Course round 1
- Teaching language English
- Application code 20122
- Open for exchange studentsYes
Credit distribution
Module | Sp1 | Sp2 | Sp3 | Sp4 | Summer | Not Sp | Examination dates |
|---|---|---|---|---|---|---|---|
| 0107 Examination 7.5 c Grading: TH | 7.5 c |
|
In programmes
- MPENM - ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 1 (compulsory elective)
- MPENM - ENGINEERING MATHEMATICS AND COMPUTATIONAL SCIENCE, MSC PROGR, Year 2 (elective)
Examiner
Patrik Albin- Associate Professor, Analysis and Probability Theory, Mathematical Sciences
Eligibility
General entry requirements for Master's level (second cycle)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.
Specific entry requirements
English 6 (or by other approved means with the equivalent proficiency level)Applicants enrolled in a programme at Chalmers where the course is included in the study programme are exempted from fulfilling the requirements above.
Course specific prerequisites
Probability theory from a first course in mathematical statistics.Some experience of computer programming such as, for example, basic knowledge of Matlab programming.
Mathematics corresponding to what students from quite math oriented educations such as TM (Technical Mathematics) or F (Engineering Physics) learn during their first year of studies.
Aim
Give a thorough introduction and oversight of the different kind of stochastic processes that are of the greatest importance in both applications in technical sciences and natural sciences as well as in further mathematical and mathematical statistical theory development.Learning outcomes (after completion of the course the student should be able to)
Perform basic calculations of continuous time and discrete time Fourier transforms and inverse transforms as well as basic understanding of their usage in probability theory (that is, characteristic functions) as well as in the context of stationary random processes and linear time invariant systems (that is, spectral analysis).
Describe how continuous time and discrete time Markov chains (including queueing systems) work from a principal theoretical point of view and by means of implementation of corresponding computer code (or so called pseudo code) as well as performing computational examples on these objects.
Describe the importance of dependence and independence between different stochastic process values (random variables) from a principal theoretical point of view as well as performing corresponding computational examples.
Describe the basic defining properties of stationary processes, wide (weak) sense stationary processes, Gaussian processes and martingales from a principal theoretical point of view as well as performing corresponding computational examples.
Content
Introduction to Fourier transforms (characteristic functions), convolutions and Diracs delta-function together with a review of some important concepts from multivariate probability theory. Definition of discrete time and continuous time stochastic processes together with their finite dimensional distributions, mean functions, correlation functions and covariance functions. Processes with independent stationary increments (Levy processes) and Gaussian processes (normal processes). A quite substantial treatment of discrete time and continuous time Markov chains including applications to queueing systems (that adds up to upwards half of the total course material). Discrete time and continuous time martingales. Continuity and differentiability of stochastic processes together with integration and summation of them. Spectral densities together with white noise in continuous and discrete time. Stochastic processes as inputs to and outputs from linear time invariant systems. Computer implementation of the majority of the above mentioned classes of stochastic processes.
Organisation
Lectures and tutorials.
Literature
Hwei Hsu: Probability, Random Variables, and Random Processes, 2nd Edition. Schaum's Outlines, McGraw-Hill 2010.
Geoffrey Grimmett och David Stirzaker: Probability and Random Processes, 3rd Edition. Oxford 2001.
Examination including compulsory elements
Written exam.
The course examiner may assess individual students in other ways than what is stated above if there are special reasons for doing so, for example if a student has a decision from Chalmers on educational support due to disability.