Tools of Modern Probability  fall semester 2020
Subject code: BMETE95AM33
Classes:
 Wednesday 10:1512:00, room H406 !CHANGE!
 Thursday 12:1514:00, room H607 !CHANGE!
HOMEWORK RESULTS:
The results of the homeworks and the calculated homework score can be found here. The list is anonymous, the names by substituted by a unique identifier called "ID". Each students can find her/his "ID" in Moodle. (Sorry for this inconvenience. The gradebook of Moodle is not well suited for such calculations.) The corrected homeworks (which were not handed out on paper) can also be viewed in Moodle. If something is missing or you find errors, please let me know!
Solutions of homework exercises are here.
END OF THE SEMESTER:
After going online, I was late with everything  sorry. So there are no more obligatory homeworks: the HW score (out of 40) is calculated from the 13 homeworks (+1 bonus exercise) that were due before 05 November. However, for those who are not satisfied with their score, there is a possibility to submit a 4th set of homeworks: 3.7; 3.9/d,f,i,k; 3.10, 3.16, 3.17. These can be submitted in Moodle, the deadline is the day before the exam (so it depends on which exam you take).
QUESTION SESSIONS:
There will be question/answer sessions before each exam via MS Teams. You can join through this link. Please come and ask, even if you are not taking the next exam: everybody is welcome.
Question session times:
 Monday, 04 January 2021, 10:0011:00
 Monday, 11 January 2021, 10:0011:00
 Monday, 18 January 2021, 10:0011:00
EXAMS:
The exam will be ORAL, via MS Teams. Please register for the exam in Neptun. When the registration closes (at noon before the day of the exam), I will decide on the order of the registered students, and let you know when it will be your turn. This may be up to 5 hours after the start of the exam. If your time is limited, please let me know, so that I can take it into account. When your time comes, log in to Teams, and I will call you in private. Also: I'm flexible about exam times. If the exams announced in Neptun are not good for you, contact me, and we try to find a suitable time.
On the exam, every student gets at least one calculation exercise (similar to those in the exercise sheets), and two randomly chosen theoretical questions from the following list. NOTE: Before the exam, everybody can choose two of these questions, which she/he would not like, to exclude from the list.
 Gaussian integrals; spherically symmetric integrals, surface of hyperspheres [notes/01, section 12]
 Almost Gaussian integrals, Laplace's method [notes/01, section 3]
 Euler gamma function, Stirling's approximation [notes/01, section 45]
 Measure space, probability space. Pushforward of measures. Distribution of random variables [notes/02, section 1.1, 1.2; 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. [notes/02, section 1.3.1; D, section 1.6]
 Construction of random variables with a given distribution, generalized inverse of distribution functions [D, Theorem 1.2.2]
 Exchanging the integral and the limit: monotone convergence theorem, dominated convergence theorem, Fatou lemma. [notes/02, section 1.3.2; D, section 1.6.2]
 Characteristic functions of random variables, characteristic functions of probability distributions. Continuity, differentiability, continuous differentiability of the characteristic function. [Exercises 2.10,...,2.14]
 Product space, product measure. Exchanging integrals: Fubini's theorem [D, section 1.7]
 Weak convergence of random variables, weak convergence of probability distributions. Equivalent descriptions, relation to strong convergence. [D, Theorem 3.2.2, Theorem 3.2.3]
 Subsequential limits for distribution functions, vague limits, tightness. [D, Theorem 3.2.6, Theorem 3.2.7]
 Relation of weak convergence and characteristic functions. [D, section 3.3.2, without proof]
 Hilbert spaces  completeness of L^2 [notes/10,11,12]
 Riesz representation theorem [notes/13]
 Absoulte continuity. RadonNikodym theorem. [notes/14]
 Conditional expectation of random variables. Definition, existence, uniqueness [notes/15]
Calculating the homework score
 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 10 homeworks are taken into account.
 Eventually, the score is scaled to be between 0 and 40 (a multiplication by 4).
ONLINE TEACHING, VERSION 2: I'll try writing detailed lecture notes and recording videos offline and publish them for you to download or watch online.
Download the notes, and have them at hand while watching the video!
In case this web page is not working (or you don't like it), all lecture notes and videos can also be found here.
Instead of live lectures, I will be available for questions on Teams on Wednesday 11:3012:00 and Thursday 13:3014:00. (If there are many questions, we can make it longer.)
Topics discussed this way:
 Linear spaces, linear transformations (definition, examples). See
 Inner product spaces, normed spaces (definition, examples). See
 Banach spaces, Hilbert spaces (definitions, example). See
 Riesz representation theorem. See
 RadonNikodym theorem. See
 Conditional expectation. See
HOMEWORK SUBMISSION: From now on, homeworks should be submitted electronically by uploading to the Moodle server of the course. Every student has an account, and hopefully everybody got their password notification emails. For each homework to submit, an "assignment" will be created in the Moodle course. Corrected homeworks will be available for every student at the same place.
Homework and practice exercise sheets are here.
HW to hand in by 01 October 2020: 1.2, 1.9, 1.11, 1.14. Bonus: 1.1
HW to hand in by 29 October 2020: 1.16, 2.2, 2.4, 2.5, 2.7.
HW to hand in by 05 November 2020: 2.9/a,c, 2.14 e or f; 2.15, 2.16.
voluntary HW to hand in by the exam: 3.7; 3.9/d,f,i,k; 3.10, 3.16, 3.17.
Videos of lectures that were live:
These videos are not cut at the natural boundaries of the material presented, but at the brakes of the classes where they were recorded. So their descriptions only describe the content roughly.
Lecture 1, part 1: Gaussian integrals (46:21, 357 MB)
Lecture 1, part 2: Spherically symmetric integrals (32:57, 263 MB)
Lecture 2, part 1: Surface of hyperspheres (45:19, 301 MB)
Lecture 2, part 2: Euler Gamma function (32:41, 134 MB)
Lecture 3, part 1: Notation for asymptotic behaviour of functions SORRY, no video: I used the wrong settings, and the quality is unacceptable.
Lecture 3, part 2: Almost Gaussian integrals 1 (49:15, 226 MB)
Lecture 4, part 1: Almost Gaussian integrals 2 (35:14, 134 MB)
Lecture 4, part 2: liminf/limsup and Almost Gaussian integrals 3 (45:14, 200 MB)
Question session (no official class on sports day): Integration of spherically symmetric functions repeated (1:02:00, 323 MB)
Lecture 5, part 1: Examples of almost Gaussian integrals (52:40, 368 MB)
Lecture 5, part 2: Normalizing almost Gaussian functions to Gaussian; Stirling's approximation (27:18, 164 MB)
Lecture 6, part 1: Beta function (36:40, 138 MB)
Lecture 6, part 2: Gamma and beta distributions in Statistical Physics (43:34, 164 MB)
Lecture 7, part 1: Measure theory basics 1 (44:00, 145 MB)
Lecture 7, part 2: Measure theory basics 2 (36:36, 117 MB)
Lecture 8, part 1: Sorry, no video  I managed to record the break instead.
Lecture 8, part 2: Distributions of random variables (33:33, 92 MB)
Lecture 9, part 1: Remarks on homework solutions (20:44, 62 MB)
Lecture 9, part 2: The notion of Lebesgue integral (1:00:40, 201 MB)
Lecture 10, part 1: Expectation of random variables (46:05, 166 MB)
Lecture 10, part 2: Density of measures and calculating integrals (32:59, 114 MB)
Lecture 11, part 1: Integral substitution theorem, expectation of distributions (43:17, 145 MB)
Lecture 11, part 2: Properties of the integral and expectation, with examples (38:26, 125 MB)
Lecture 12, part 1: Description and construction of random variables (51:37, 147 MB)
Lecture 12, part 2: Generalized inverse of distribution funcitons, remark about the Zikkurat method (31:35, 81 MB)
Lecture 13, part 1: Types of convergence of functions and random variables (42:58, 135 MB)
Lecture 13, part 2: Convergence of functions vs convergence of integrals (36:09, 107 MB)
Lecture 14, part 1: Exchanging limit and integral: Monotone concvergence thm, Dominated convergence thm, Fatou's lemma (53:29, 153 MB)
Lecture 14, part 2: Example of dominated convergence: continuity and differentiability of characteristic functions (31:29, 85 MB)
Lecture 15, part 1: Differentiability of the characteristic function continued (38:39, 130 MB)
Lecture 15, part 1: Exchanging integrals: Fubini's theorem (34:49, 119 MB)
Lecture 16, part 1: Weak convergence of random variables, distributions and distribution functions (41:06, 158 MB)
Lecture 16, part 2: Equivalence of notions of weak convergence; relation to strong convergence (35:00, 128 MB)
Lecture 17, part 1: Weak convergence on metric spaces, Skorokhod representation theorem (33:28, 113 MB)
Lecture 17, part 2: Subsequential limits for distribution functions, vague limits, tightness (52:20, 180 MB)
Lecture 18, part 1: Weak convergence and tightness (27:34, 108 MB)
Lecture 18, part 2: Why we like limit theorems, weak convergence and characteristic functions (38:24, 152 MB)
ERROR REPORT REQUEST: The lecture notes and videos certainly contain errors. I am grateful to anyone finding and reporting these errors. Please let me know of every error you find!
Lecturer: Imre Péter Tóth
Course organization:
 There will be lectures, homeworks and an oral final exam. The homeworks contribute to the grade with 40% weight.
 Lectures will be held at the university as long as this is possible. I suggest attending the lectures for everybody who can, but there is no obligation.
 Detailed lecture notes and lecture videos will be available online, on the course home page.
 Homeworks will be published electronically on the coure home page, and must be submitted electronically.
 Results of the homeworks will be published on the course home page (in anonym form), and corrected homeworks will be sent back to the students electronically.
 There will be regular question sessions online (via video conference using MS Teams) for everybody.
 The lectures consist of theoretical and practice parts, but these will not be strictly separated in time.
 Students can take the final oral exam online (via video call using MS Teams) or personally, if the university regulations will make this possible.
Grading rules:
 Over the course, the maximum total score is 100: 40 for the homeworks and 60 for the final oral exam.
 The students get homeworks with a total value of 60 scores, but only the best 2/3 is taken into account.
 On the oral exam, every student will get at least one practice exercise.
 To obtain the signature, at least 50% of the homework scores  which is 20 scores  must be reached.
 To pass the exam, at least 50% of the exam scores  which is 30 scores  must be reached.

Final score to grade conversion table:
score  grade 
049  1 
5062  2 
6375  3 
7688  4 
89100  5 
Planned topics: for sure, not all of these will be covered.

Gaussian integrals and a geometric application: surface of hyperspheres. Almost Gaussian integrals, Laplace's method.
Euler gamma function, Stirling's formula.

Measure theoretic foundations of Probability theory: integral and expectation; pushforward of measures and distribution of random variables; integral substitution and expectation of distributions. Density of measures and random variables.
Exchangeablilty of expectation and limit: monotone and dominated convergence theorems, Fatou's lemma. Fubini's theorem.

Weak convergence of random variables and distributions. Construction of a random variable with a given distribution. Relation of strong and weak convergence. Existence of weak limits  Cantor's diagonal argument, tightness: Helly's theorem.

Hilbert spaces, Riesz representation theorem. L^2 spaces: completeness. Absolute continuity, RadonNikodym theorem.

Fourier series and Fourier transform, point spectrum and continuous spectrum. Spectral theorem and Gelfand's spectral radius formula.

Conditional expectation of random variables  existence, uniqueness. Jensen's inequality for conditional expectations.

Basics of normed spaces: spaces of continuous functions; measures as functionals (weak* topology), ArselaAscoli theorem.
Hölder's inequality, HahnBanach theorem, BanachAlaoglu theorem.

A few basic partial differential equations: wave equation, heat equation. Laplace operator, harmonic functions. Relation with complex differentiability in two dimensions.

Conformal mappings, conformal equivalence of domains. Riemann mapping theorem.
Application of linear fractional transformations in solving the Laplace equation. Discrete Laplace operator and equation: hitting probabilities for random walks and the Wiener process.
Suggested literature
 [TIP] Draft lecture notes written by the lecturer: notes/.
 [D] R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
 [R2] Rudin: Real and Complex Analysis