- Study properties of probability spaces and independently prove simple statements;
- model random experiments using probability spaces and random variables;
- Know some examples for distributions, e.g., uniform, binomial, Poisson, exponential and normal distribution;
- Describe distribution using distribution functions and densities;
- Apply conditional probability and Bayes' theorem;
- Inspect events and random variables for stochastic independence;
- Understand moments of random variables and their calculation rules, e.g., for the expectation and variance of random variables;
- Know multi-dimensional random variables, e.g., the normal distribution in R^n, and determine marginal distributions;
- Compute covariance and correlation coefficients of multi-dimensional random variables;
- Know stochastic convergence concepts and their interrelations;
- Understand and apply the law of large numbers, e.g., for estimators for the expectation and variance of random variables in statistics;
- Work with moment generating and characteristic functions and their properties;
- Comprehend and apply the central limit theorem.
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Sample space, events, probability space and random variables, probability distributions and densities, conditional probability, stochastic independence, multi-dimensional random variables, moments, covariance, correlation, types of stochastic convergence, laws of large numbers, moment generating and characteristic functions, the central limit theorem
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