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 # when topic remark week 1-2 2017.09.19,26 Review of basic notions of probability theory. Measure-theoretic language. Some famous problems and paradoxes. week 3 2017.10.03 Different types of convergence for random variables. Borel-Cantelli lemmas. week 4-5 2017.10.10,17 Laws of Large Numbers. The method of characteristic functions in proving weak convergence: the Central Limit Theorem. week 6-7 2017.10.24,31 Conditional expectation with respect to a sub-sigma-algebra. Martingales. Some martingale convergence and optional stopping theorems. week 8-9 2017.11.07,14 Applications of martingales: Galton-Watson branching processes. Asymptotic results. Birth and death process. ABRACADABRA problem. week 10-11 2017.11.21,28 Random walks on the integers. Construction and basic properties of Brownian motion. week 12 2017.12.05 Probabilistic 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.

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:
• 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.
• The midterm is worth 30 points, and the final is worth 40 points. These are added to the homework score to give the "total score".
• The total score is translated into grades using the following table:  score range grade 0-49 F 50-59 C+ 60-67 B- 68-75 B 76-83 B+ 84-89 A- 90-100 A

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