Tools of Modern Probability -- fall semester 2019

Subject code: BMETE95AM33

Classes: Wednesday 16:15-18:00 and Thursday 12:15-14:00, room H45/A.

Lecturer: Imre Péter Tóth

QUESTIONS/ANSWERS SESSION before the third exam: Monday, 27 January 2020 from a 11:00 in room H46. Everybody welcome!

RESULTS: here.
On the exams, the best 5 exercises are taken into account. On the first exam, all exercises were graded on a scale of 10. To get a maximum score of 70, the sum of these scores was multiplied by 1.4 to give the "Exam_1_SCORE corrected" version. On the second exam, all exercises were graded on a scale of 14.

The "ID" in the first column is a unique identifiter for each student, that I wrote into Neptun as a score for a nominal exercise titled "unique identifier". This way everybody can find the results of their own, but hopefully not of the others. I hope this satisfies all existing data protection regulations.

Homework sheets: here.

Homeworks to hand in by 09 October: 1.4; 1.9; 1.10
Homeworks to hand in by 31 October: 2.3/b; 2.7; 2.8
Homeworks to hand in by 21 November: 2.9/II,III,IV; 2.15; 2.16; 3.2
Homeworks to hand in by 11 December: 3.7; 4.1; 4.8; 4.15; 4.16

Sample exam exercise sheet, updated for fall 2019: here.

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.
• [R2] Rudin: Real and Complex Analysis

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. [Exercises 2.10,...,2.13]
• 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, properties. [D, section 5.1]
• Jensen's inequality for conditional expectations. [D, Thm 5.1.3]
• Independence of random variables, independence of sigma-algebras. [D, section 2.1 (first 2 pages), section 2.1.2]
• Construction of random variables with a given distribution. [D, Theorem 1.2.2]
• Weak convergence of random variables, weak convergence of probability distributions. Equivalent descriptions, realtion to strong convergence. [D, Theorem 3.2.2, Theorem 3.2.3]