(prob:intro)=
# Introduction to probability theory

Probability theory is an axiomatic approach to probability, assigning 

```{dropdown} Stochastic variables
- Definition of stochastic variable
- Discrete and continuous stochastic variables
  - Probability functions, moments (if they exists, see heavy-tailed distribution), and examples
- Multi-dimensional stochastic variables:
  - joint, conditional, marginal probability
  - Bayes' theorem
  - independence
  - moments: covariance, correlation

- Generators...
- I.i.d. variables: law of large numbers, central limit theorem; convergence of statistics (reference to measure in the definition of a sthocastic variable)
- Sampling

- Extra:
  - heavy tails probability functions

```

```{dropdown} Stochastic processes
- Definition of stochastic process
- Time-continuous/time-discrete
- Ergodicity and stationariety:
  - moments, correlation,...
  - analysis in time and Fourier domains of time-signals
- Applications:
  - example of processes:
    - white noise
    - Wiener process (Brownian motion): definition, application, relation with 
    - discrete-time Markov process (useful in RL, can be interpreted as a discretized continuous process)
  - response of LTI to random input

```

```{dropdown} Stochastic fields
```