### 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-NikodÃ½m 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.

#### Resources

- Lecture Notes: Probability Theory I (in German)

#### Simulations

wuerfel.py: Relative frequencies in a dice game.

ausfall.py: Probability of default of a camera.

bernoulli.py: Bernoulli distribution.

binomial.py: Binomial distribution.

geometrisch.py: Geometric distribution.

hypergeometrisch.py: Hypergeometric distribution.

poisson.py: Poisson distribution.

poisson_approx.py: Poisson approximation.

uniform.py: Uniform distribution.

exponential.py: Exponential distribution.

normal.py: Normal distribution.

ggz.py: Law of large numbers.

zgs_pfad.py: Central limit theorem (single realization).

zgs_verteilung.py: Central limit theorem (histogramm).