Tools of Modern Probability  fall semester 2018
Subject code: BMETE95AM33
Classes: Tuesday 10:1512:00, room T605 and Wednesday 10:1512:00, room T605.
Lecturer: Imre Péter Tóth
TIME CHANGE for the 3rd exam: the exam will start at 12:00 (on Tuesday, 15 January 2018).
QUESTIONS/ANSWERS session
 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 MoivreLaplace CLT written by the lecturer: De MoivreLaplace 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 45]

Measure space, probability space. Pushforward of measures. Distribution of random variables [TIP, section 6.1, 6.2; D, section 1.11.5]

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. RadonNikodym 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. [??]
Grading rules:

There are/will be homeworks and a written final exam. The homeworks contribute to the grade with 30% weight. On the exam, I will give at least one practice exercise.

Calculating the partial grade for the homeworks:
 The solution to each homework is evaluated with a "code" with the following meaning:
 "3" means correctly solved
 "2" means solved with some error
 "1" means started on a correct track, but not solved
 "0" means completely wrong.
At the end of the semester these codes are translated to a homework score. At the translation, the big difference is between "solved" and "not solved", so a "correctly solved" is worth 1 point, a "solved with some error" is worth 0.8 points, and the rest is worth no points at all.

Only the best 70% of the homeworks are taken into account.