Tools of Modern Probability -- fall semester 2018

Subject code: BMETE95AM33

Classes: Tuesday 10:15-12:00, room T605 and Wednesday 10:15-12:00, room T605.

Lecturer: Imre Péter Tóth

TIME CHANGE for the 3rd exam: the exam will start at 12:00 (on Tuesday, 15 January 2018).

• before the 3rd exam: 14 January 2019 (Monday) 17:00, room H46. (Sorry, I can't make it earlier.)
• before the 2nd exam: 07 January 2019 (Monday) 11:00, room H663.
• before the 1st exam: 17 December 2018 (Monday) 17:00, room H663.
Sample exam exercise sheet here.

First exam exercise sheet here. There are 7 exercises, and students had to choose 5 of these.

Homework sheets: here.

Homeworks to hand in by 26 September: 1.2; 1.5; 1.9; 1.10
Homeworks to hand in by 17 October: 2.5; 2.14; 2.15; 2.16
Homeworks to hand in by 24 October: 3.3; 3.4; 3.6
Homeworks to hand in by 13 November: 3.8/c,d,f,h,i,j; 4.1; 4.2; 4.14

Suggested literature
Most of the material discussed in class is covered in the following literature. From the books, you only need a tiny bit, detailed below.

• [TIP] Draft lecture notes written by the lecturer: Tools of Modern Probability.
• [CLT] Proof of the De Moivre-Laplace CLT written by the lecturer: De Moivre-Laplace CLT.
• [D] R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
• [R1] Rudin: Principles of mathematical analysis
• [R2] Rudin: Real and Complex Analysis
• [R3] Rudin: Functional Analysis
• [R4] Rudin: Fourier Analysis on Groups

What actually was covered: detailed list with references
• Gaussian integrals [TIP, section 1]
• Polar coordinates in higher dimensions, surface of hyperspheres [TIP, section 2]
• Almost Gaussian integrals, Laplace's method [TIP, section 3]
• Euler gamma function, Stirling's approximation [TIP, section 4-5]
• Measure space, probability space. Push-forward of measures. Distribution of random variables [TIP, section 6.1, 6.2; D, section 1.1-1.5]
• Integral, expectation. Integration by substitution. Expectation of random variables. Densities of measures. Sums of series and Riemannian integrals as special cases of the (Lebesgue) integral. [TIP, section 6.3.1; D, section 1.6]
• Charactersitic functions of random variables, characteristic functions of probability distributions.
• Exchanging the integral and the limit: monotone convergence theorem, dominated convergence theorem, Fatou lemma. [TIP, section 6.3.2; D, section 1.6.2]
• Application: continuity of the characteristic function. [Exercise 2.14]
• Product space, product measure. Exchanging integrals: Fubini's theorem [D, section 1.7]
• Hilbert spaces - Riesz representation theorem [R2, Chapter 4]
• Absoulte continuity. Radon-Nikodym theorem. [TIP, Def. 6.39, Thm 6.40; R2, Theorem 6.10]
• Conditional expectation of random variables. Existence, uniqueness. [D, section 5.1]
• Jensen's inequality for conditional expectations. [D, Thm 5.1.3]
• Fourier series [R1, Chapter 8; R2, Chapter 4, 2nd and 3rd section]
• Fourier transform, Plancherel's theorem [R2, Chapter 9]
• Laplace operator, harmonic functions. Relation with complex differentiability in two dimensions. Conformal mappings, conformal equivalence of domains. Application of linear fractional transformations in solving the Laplace equation. [??]