Probability 1 at CEU -- fall semester 2017

This was the midterm, and here are the solutions.

Homework sheet 1. Solutions here.
Homework sheet 2. Solutions here.
Homework sheet 3. Solutions here.
Homework sheet 4. A typo in Exercise 4.8 corrected - thanks to Rumen. Solutions here.
Homework sheet 5. Solutions here.
Homework sheet 6. Solutions here.

No. of Credits: 3
No. of ECTS credits: 6
Time Period of the course: Fall Semester
Prerequisites: basic probability
Course Level: introductory MS
Official syllabus: here.
Classes: Tuesdays from 12:30 in room 310/A.
Course Coordinator: Imre Péter Tóth.

Schedule (planned):
week #whentopicremark
week 1-22017.09.19,26Review of basic notions of probability theory. Measure-theoretic language. Some famous problems and paradoxes.
week 32017.10.03Different types of convergence for random variables. Borel-Cantelli lemmas.
week 4-52017.10.10,17Laws of Large Numbers. The method of characteristic functions in proving weak convergence: the Central Limit Theorem.
week 6-72017.10.24,31Conditional expectation with respect to a sub-sigma-algebra. Martingales. Some martingale convergence and optional stopping theorems.
week 8-92017.11.07,14Applications of martingales: Galton-Watson branching processes. Asymptotic results. Birth and death process. ABRACADABRA problem.
week 10-112017.11.21,28Random walks on the integers. Construction and basic properties of Brownian motion.
week 122017.12.05Probabilistic methods in combinatorics - an example.

Midterm exam:
will be on Friday, 10.Nov.2017 at 14:00.

Suggested literature (References):
  1. To catch up with the missing basics: W. Feller: An Introduction to Probability Theory and Its Applications, Vol. 1, 3rd Edition, Wiley, 1968.
  2. For most of the course: R. Durrett: Probability. Theory and Examples. 4th edition, Cambridge University Press, 2010.
  3. For the short combinatorics part: N. Alon, J. H. Spencer: The Probabilistic Method. 3rd edition, Wiley, 2008.

Grading rules:
There will be weekly homework assignments wort 30% of the total score. The midterm will also be worth 30%, while the final exam is worth 40%.
In detail:

Material for learning:
Some notes about the measure theoretic basics of Probability