Probability 1 at CEU -- fall semester 2014

THE FINAL EXAM will be oral, on the 15th of December 2014 between 9 and 11 am. Here is a fairly detailed list of what to know.
Every student will get two questions at random from this list.

All results of the semester are / will be here for those who request that I publish them online.

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

Material for learning at the bottom of the page

Homework sheet 1. Solutions here.
Homework sheet 2. Solutions here.
Homework sheet 3. Solutions here.
Homework sheet 4. Solutions here.
Homework sheet 5. Solutions here.
Homework sheet 6. Solutions here.
Homework sheet 7, Solutions here.
Homework sheet 8. 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
Syllabus: here. Classes: Thursdays from 12:30 in room 301.
Course Coordinator: Imre Péter Tóth.

Schedule (planned):

week #	when	topic	remark
week 1-2	2014.09.25, 10.02	Review of basic notions of probability theory. Measure-theoretic language. Some famous problems and paradoxes.
week 3	2014.10.09	Different types of convergence for random variables. Borel-Cantelli lemmas.
week 4	2014.10.16	Laws of Large Numbers. The method of characteristic functions in proving weak convergence: the Central Limit Theorem.
week 5-6	2014.10.21,30	Conditional expectation with respect to a sub-sigma-algebra. Martingales. Some martingale convergence and optional stopping theorems.	23 October is a public holiday. Instead of it, the class will be on Tuesday, the 21st of October at 13:00
week 7	2014.11.06	Applications of martingales: Galton-Watson branching processes. Asymptotic results. Birth and death process.
week 8-9	2014.11.13,20	Probabilistic methods in combinatorics. Second moment method, Lovász Local Lemma.
week 10	2014.11.27	Some large deviation theorems, Azuma's inequality.
week 11-12	2014.12.04,11	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: 04 November 2014 (Tuesday) at 13:00
Final exam: Preliminary plan: 12 December 2014, 9:00 Not at all sure to be good.

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:

• 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 year, mostly with solutions