Probability 1 at CEU -- fall semester 2015

Here is a sample of possible questions for the final exam.

Homework sheet 1
Homework sheet 2. Solutions here.
Homework sheet 3. Solutions here.
Homework sheet 4. Solutions here, except for exercise 4.9, which is here.
Homework sheet 5 (voluntary). Solutions here.

This was the midterm exam, and here are the solutions.
This was the final exam, and here are the solutions.

No. of Credits: 3
No. of ECTS credits: 6
Time Period of the course: Fall Semester
Prerequisites: basic probability
Course Level: introductory MS
Syllabus: here. Classes: Thursdays from 12:30 in room 301.
Course Coordinator: Imre Péter Tóth.

Schedule (planned):

week #	when	topic	remark
week 1-2	2015.09.22,29	Review of basic notions of probability theory. Measure-theoretic language. Some famous problems and paradoxes.
week 3	2015.10.06	Different types of convergence for random variables. Borel-Cantelli lemmas.
week 4	2015.10.13	Laws of Large Numbers. The method of characteristic functions in proving weak convergence: the Central Limit Theorem.
week 5-6	2015.10.20,21	Conditional expectation with respect to a sub-sigma-algebra. Martingales. Some martingale convergence and optional stopping theorems.	There will be no class on 27 October. Instead of it, the class will be on Wednesday, the 21st of October at 14:00 in room 301.
week 7	2015.11.03	Applications of martingales: Galton-Watson branching processes. Asymptotic results. Birth and death process.
week 8-9	2015.11.10,17	Probabilistic methods in combinatorics. Second moment method, Lovász Local Lemma.
week 10	2015.11.24	Some large deviation theorems, Azuma's inequality.
week 11-12	2015.12.01,08	Random walks on the integers. Construction and basic properties of Brownian motion.

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.

Midterm, exam:
Midterm: 06 November 2015 (Friday) at 12:30, room 301.
Final exam: 11 December 2015 (Friday) at 12:30, room 301.

Grading rules:
There will be weekly homework assignments worth 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 will be 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 will be 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 will be added to the homework score to give the "total score".
• The total score will be 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

Requirements for "audit":
regular participation in class and a short oral account of the concepts and phenomena learned.

Material for learning:
Some notes about the measure theoretic basics of Probability
Homework sheets, midterm exams, sample exam and final exam of the previous years, mostly with solutions