Introduction to Probability and Statistics

- probability spaces, conditional probabilities, random variables, independence of events and random variables, expectation, variance, covariance, transformation formula for multivariate random variables, limit theorems.
- statistical models, estimators, statistical tests.
- introduction to R, transformation and visualisation of data in R, illustration of probabilistic concepts in R, implementation and practical comparison of statistical methods in R, communication of results of data analysis using R.

Upon completion of the module, students are able to
- understand basic models, concepts, and methods from probability theory and statistics, and formulate them in a mathematically precise way.
- discuss and prove connections between these concepts and illustrate them with examples; solve problems using these concepts and the methods covered in the class
- model simple random experiments and statistical methods and implement them in a computer program
- interpret statistical data and methods, visualize data, and assess the meaning of random experiments

MA0001 - Analysis 1
MA0002 - Analysis 2
MA0004 - Linear Algebra 1
MA0005 - Linear Algebra 2 und Discrete Structures

The course is offered as lecture with accompanying exercise session and a supplement focusing on practical applications. During the lecture, the material will be presented using illustrative examples and mathematical proofs. The lecture intends to motivate the students to look into the subject and to study the literature themselves. The practical supplement relates the material with an implementation with a computer using a combination of lecture and individual guided implementation with R. Exercise sheets and their solutions will be offered to the students so that they can check their work and deepen the learnt methods, concepts and strategies for practical implementation.
After this was done in the beginnung under guidance during the exercises sessions and the practical supplement, this is deepened during the semester individually and partially also in small groups.

blackboard, slides, R statistical software

Georgii, H.-O. (2007). Stochastik, De Gruyter.
Kersting, G., Wakolbinger, A. (2008). Elementare Stochastik. Birkhäuser, Basel.
Wickham, H. und Grolemund, G. (2017). R for Data Science. O'Reilly.

Weiterführende Literatur:
Grimmett, G., Stirzaker, D. (2001). Probability and Random Processes. Third Edition. Oxford University Press, Oxford.
Dehling, H., Haupt, B. (2004). Einführung in die Wahrscheinlichkeitstheorie und Statistik. 2. Auflage. Springer, Berlin.

The module examination is given as a written exam (90 minutes). Based on questions about knowledge and comprehension, it is verified whether the students
- can formulate basic models and concepts from probability and statistics in a mathematically precise way and use them accurately,
- can model simple random experiments and statistical problems, understand R programs and interpret their output,
- can interpret statistical data and evaluate the outcome of random experiments.