Tools of Modern Probability  fall semester 2019
Subject code: BMETE95AM33
Classes: Wednesday 16:1518:00 and Thursday 12:1514:00, 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!
The first exam exercise sheet was this.
Solutions are here.
The second exam exercise sheet was this.
Solutions are here.
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 MoivreLaplace CLT written by the lecturer: De MoivreLaplace 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 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. [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. RadonNikodym 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 sigmaalgebras. [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]
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.

Final score to grade conversion table:
score  grade 
039  1 
4054  2 
5569  3 
7084  4 
85100  5 