Probability Theory I

The lecture covers the following topics:

Foundations of Probability Theory
Why we study probability theory, modeling random experiments, rules of calculus for probability measures, abstract probability spaces and random variables, examples of distributions.

Real-Valued Random Variables
Cumulative distribution functions of real-valued random variables, densities and the Radon-Nikodým Theorem, expectation, variance, and correlation, characteristic functions and higher-order moments.

Conditional Probability and Independence
Conditional probability, independence of events and random variables, product spaces and joint distributions, joint distribution of independent random variables, sums of independent random variables and convolutions.

Types of Convergence and Fundamental Limit Theorems
The weak law of large numbers, convergence in probability and almost sure convergence, the strong law of large numbers, convergence in distribution, the central limit theorem.


Simulations Relative frequencies in a dice game. Probability of default of a camera. Bernoulli distribution. Binomial distribution. Geometric distribution. Hypergeometric distribution. Poisson distribution. Poisson approximation. Uniform distribution. Exponential distribution. Normal distribution. Law of large numbers. Central limit theorem (single realization). Central limit theorem (histogramm).