Probability 1 at CEU -- fall semester 2018

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.

---

A few midterm exercises from earlier years 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
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	2018.09.17,24	Review of basic notions of probability theory. Measure-theoretic language. Some famous problems and paradoxes.
week 3	2018.10.01	Different types of convergence for random variables. Borel-Cantelli lemmas.
week 4-5	2018.10.08,15	Laws of Large Numbers. The method of characteristic functions in proving weak convergence: the Central Limit Theorem.
week 6-7	2018.10.25,29	Conditional expectation with respect to a sub-sigma-algebra. Martingales. Some martingale convergence and optional stopping theorems.	Oct 22 is a holiday, the class will be moved to the 25th, 09:00.
week 8-9	2018.11.05,12	Applications of martingales: Galton-Watson branching processes. Asymptotic results. Birth and death process. ABRACADABRA problem.
week 10-11	2018.11.19,26	Random walks on the integers. Construction and basic properties of Brownian motion.
week 12	2018.12.03	Probabilistic methods in combinatorics - an example.

Midterm exam:
will be on Wednesday, 31.Oct.2018 at 16:30, room 310/A

Final exam:
will be on Monday, 10.Dec.2018 at 09:00, room 310/A

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 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 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