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.

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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.

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 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