ANALYTICAL AND STOCHASTIC MATHEMATICAL METHODS FOR ENGINEERING

Analytical Methods

1. Lebesgue Measure and Integral
2. Elements of Abstract Measure and Integral
3. Normed Spaces and Banach spaces
4. Inner product spaces, Hilbert spaces
5. Elements of Harmonic Analysis: Fourier series and integrals.

Probabilistic Methods

6. Introduction to probability — probability spaces, axioms of probability, conditional probabilities, independence of events.
7. Random variables (discrete and continuous) — definition, expectation and moments. Examples of random variables and applications, with a focus on Gaussian random variables.
8. Random vectors.
9. Characteristic function.
10. Convergence of random variables: weak, in probability, in L^p, almost sure.
11. The law of large numbers and the central limit theorem with applications.
12. Conditional expectation.
13. Martingales in discrete time.