VU 325.062 Stochastics
Stochastics is a sub-area of mathematics and, as a generic term, combines the areas of probability theory and statistics. In practically all areas of technology and natural sciences, chance occurs as a mostly undesirable factor. In production, manufacturing technology, quality assurance and in general when processing measured data, random phenomena must be methodically processed correctly. On the one hand, representative key figures must be determined from existing data (statistics), and on the other hand, statements about future measurement results should be possible on the basis of such random samples (probability theory - stochastics). The stochastics course is intended to impart solid knowledge of the statistical and stochastic basics, such as probability, random variables, distribution density functions and expected value. In addition, the problem of parameter estimation from random samples is presented together with hypotheses and tests. This should enable a basic understanding of the connections between chance, probability and prediction. The processing of measured data can thus be carried out correctly in terms of methodology, and the range of validity for statements about the underlying processes can be reliably specified. It is also essential to be able to estimate models and their parameters from data and to correctly state the uncertainties of the parameters quantitatively. This means that even complex regression models can be reliably created. Overall, this conveys the basic skills with regard to random phenomena, which are indispensable in modern engineering sciences but also in everyday work.
In the first section of the course, distributions and expected values, probability, definition of a random variable, strict and weak description, confidence interval and the interaction of several random variables are dealt with. This subject area is often abstract and theoretical in nature, but represents the important foundation for understanding the following chapters. After these basic definitions, an already practically important part is taught: the calculation of probabilities, how to estimate values of statistical parameters (e.g. mean and variance ) and that these parameters only make sense in combination with confidence intervals. The central limit theorem is the basis and is also explained here. A large section is dedicated to the hypothesis tests, since these embody the scientific methodology on the one hand and answer many practice-relevant questions aptly on the other. The core of a hypothesis test is a random sample (measurement, sample), which is the basis for a general statement about the population. The possible errors of the first and second kind, one-sided and two-sided tests, as well as tests for one or two samples are dealt with. A separate section is dedicated to analysis of variance (ANOVA). Another important method is regression analysis, which allows models with linear parameters to be optimally fitted to measured data. The basic methods of least squares estimation, their properties and hypothesis tests for the model parameters are taught. The residual analysis as a simple model validation is also presented. The graduates of the course should on the one hand be able to solve fundamental problems in stochastics and on the other hand have acquired a solid foundation for independent in-depth study in related subject areas.