Probability Theory - Spring 2005

Benedek Valkó

Alfréd Rényi Institute of Mathematics,	Technical University Budapest
Hungarian Academy of Sciences	Mathematical Institute, Department of Stochastics
H-1053 Budapest, Reáltanoda u. 13-15.	H-1111 Budapest, Egry J. u. 1.
Phone: 483-8300	463-1101
Email: valko-at-renyi.hu	valko-at-math.bme.hu

Classes are held on Monday 10:00-12:00 and on Wednesday 8:00-9:00 in Room 206. 

Office Hour: Wednesday 9:00-10:00, Room 206.

The final exam will be held on May 23 (Monday) 10:00-12:00 in Room 206.

Covered Topics (so far)   

Handout on continuous random variables: pdf

Handout on geometric probability problems and general probability spaces:          pdf

Problem set #1 (pdf)
Due date: 2. 16
Solutions (pdf)
Problem set #2 (pdf)
Due date: 2. 23
Solutions (pdf)
Problem set #3 (pdf)
Due date: 3. 2
Solutions (pdf)
Problem set #4 (pdf)
Due date: 3. 9
Solutions (pdf)
Problem set #5 (pdf)
Due date: 3. 19
Solutions (pdf)
Midterm exam (pdf)
3. 19
 
Problem set #6 (pdf)
Due date: 4. 4
Solutions (pdf)
Problem set #7 (pdf)
Due date: 4. 11
Solutions (pdf)
Problem set #8 (pdf)
Due date: 4. 18
Solutions (pdf)
Problem set #9 (pdf)
Due date: 4. 27
Solutions (pdf)
Problem set #10 (pdf)
Due date: 5.9
Solutions (pdf)
Problem set #11 (pdf)
Due date: 5.18
         
Solutions (pdf)   

Approximate values of the standard normal distribution function: txt

You can find the course syllabus here.

Some of the ,,highlights'':

During the course we will answer the following questions (beside others):

• Is it really possible to predict the outcome of an election with high precision just by making a poll with a couple of thousand voters?       
• We start flipping a coin and after each coin toss we write down an H if the # of heads is more than the # of tails up to this point (otherwise we don't do anything). After a 1000 coin tosses we see that we have written down more than 900 H's. Is it reasonable to believe that the coin we used is biased?
• Why is it true that a drunken man will eventually find his way home, but a drunken bird may be lost forever? (And what does that mean?)

• Suppose that a type of amoeba reproduces itself the following way: after a minute it will die with a probability of 1/3, splits into two with a probability of 1/3 or splits into three with a probability of 1/3. We start an amoeba culture by placing one amoeba in a petri dish. What is the probability that the colony will eventually die out?

We also show that probability theory may be used to prove non-trivial results for other fields of mathematics, e.g. combinatorics, number theory or approximation theory.

Grading and assignments: There will be weekly homework assignments to be handed in during the semester (50% of the final grade), a midterm exam (20%), and a final exam (30%).

• Homework assignments: Every week you will receive 5 problems which are due for the next week. A correct solution will be worth 3 points (it is possible to receive a fraction of this), each weekly assignment is worth a maximum of 15 points. At the end of the semester the sum of your 10 best weekly results will give your total homework points (50% of your final grade). There will be bonus problems which may be handed in for extra credit and may increase your total homework points (but not beyond 150 points). Group work is encouraged, but write up your own solution. Please show your work leading to the result, not only the result.
• In addition to the homework assignments you will receive plenty of problems for practice. It is recommended (though not compulsory) that you try to solve as many of these as possible. I intend to discuss some of these problems during my office hour.
• The midterm exam will be held on March 19. It will be worth 60 points (20% of your final grade).
• The final exam will be held on the week of May 23, the exact date will be announced later. It will be worth 90 points (30% of your final grade).

There will be no (probability) classes on May 2 and 4 as I will be away on a conference. I hope to make up these classes (and maybe some of the others lost due to Easter and Whit Monday) on some Monday mornings.

I am planning to provide up-to-date information about the course on this page (covered topics, weekly assignments ...).

If you have any questions or suggestions regarding the course, feel free to contact me in person or via email.