3. Introduction to probability theory#

Probability theory is an axiomatic approach to probability, assigning

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

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

Stochastic fields