MA2505
- Probability Theory, 2011-2012
"It
is remarkable that a science which began with the consideration of
games of chance should have become the most important object of human
knowledge." (
Pierre Simon Laplace,
Théorie Analytique des Probabilités, 1812)
"Everything existing in the universe is the fruit of
chance."
(Democritus)
"I know too well that these arguments from probabilities are imposters,
and
unless great caution is observed in the use of them, they are apt to be
deceptive.
" (Plato, Phaedo)
[Contact]
[Timetable] [Course information] [Notes] [Homework]
[Links] 
[Exams]
Contact
details.
Office: Fraser Noble building Room 136
Office time: TBA
Email: a dot libman dot abdn dot ac dot uk
Timetable
Lectures:
Mon
14.00-15.00 New King's 3
Thu
13.00-14.00 New King's 3
Tutorial:
Fri
11.00-12.00 King's College T2
Course
Information
Course
description form: [pdf]
Problem sets:
These
will be handed out on a weekly basis. They are an integral part of the
course. You are encouraged to
work in groups.
Continuous assessments:
They will be handed out several times throughout the semester in a
frequency that will be determined later as I find
appropriate. Limited group-work is permitted within reason.
The rule is: "thinking
in groups: yes; writing in groups: please don't".
Textbooks: My policy is to encourage students to use
the library
and find the books that suit them best. You are likely to find
everything you need in the library. Here are some books I liked:
- "Probability,
an introduction", Geoffrey Grimmett and Dominic Welsh
(Oxford University Press).
- "An
Introduction to Probability Theory and Its Applications,
Volume I", William Feller.
- "A first
course in Probability", Sheldon Ross.
All books are available in the library. I find the three
above excellent.
Departmental books:
The department has purchased several copies of the book by Ross.
At the moment they are in my possession and it is possible to loan the
book from me for a limited period.
Exam information
Vital information is here [pdf]
In the exam you will be given the following sheets: Series and distribution functions [pdf]; The normal distribution [pdf]
Past papers
2010-2011 [pdf]
Notes
from lectures and tutorials
Notes to download [pdf]
Some material that will be handed out in the exam
Power series and distribution
functions: normal font [pdf]
large print [pdf]
Normal distribution table:
normal font [pdf]
large print [pdf]
Synopsis of lectures and tutorials:
Week 1
30.01.2012 Introduction,
countable sets, density funcions.
02.02.2012 Probability
spaces, properties of the probability function.
03.02.2012 Properties of the
probability function. Elementary examples.
Week 2
06.02.2012 Uniform
probability spaces. Counting principles.
09.02.2012 Combinatorics.
10.02.2012 (Tutorial:
Questions done in class: 1,6,7,8,9)
Week 3
13.02.2012 The
inclusion-exclusion principle.
16.02.2012 Conditional probability. Conditional
probability as a probability function, the Partition Theorem.
17.02.2012 (Tutorial). Questions
3,4,6,7,8,11,12,13
Week 4
20.02.1012 The partition theorem (examples, Bayes' formula,
Marilyn's
goats), Independent events.
23.02.2012 Independent events; Bernoulli trials,
the Gambler's
Ruin problem.
24.02.2012 Tutorial
Week 5
27.02.2012 percolation
on binary trees, Random variables,
Density functions, Expectation
01.03.2012 Conditional
expectation. The partition theorem. The linearity of the expectation.
Examples.
02.03.2012 Tutorial
Week 6
05.03.2012 Independent random variable. The expectation of a
product of random variables.
08.03.2012 The variance and covariance.
09.03.2012 Tutorial
Week 7
12.03.2012 The variance and covariance.
15.03.2012 The Bernoulli
distribution and counting arguments. The binomial
distribution.
16.03.2012 Tutorial
Week 8
The negative
binomial distribution. The Poisson
distribution.
The Poisson distribution. The hypergeometric
distribution.
An
important remark (mainly to myself): In the lectures I forgot to
calculate the expectation and the variance of random variables with the
Poisson distribution. This will be done later on when we will study
generating functions. For the time being you may find E(X) and Var(X)
in any textbook or on the web.
Week 9
Markov's inequality. Chebyshev's inequality. The
weak law of large numbers.
The weak law of large numbers. The central
limit theorem.
Table of the normal distribution [pdf]
Week 10
The central limit theorem. Generating function.
Generating functions.
Week 11
Generating functions. Branching processes.
Branching processes: a model for population growth
Week 12
Branching processes: a model for population growth.
A model for population growth. Markov chains.
Homework
Week 1: Problem set 1
(03 Feb. 2012).
Questions [pdf]
Solutions
[pdf]
Week 2: Problem set 2
(10 Feb. 2012).
Questions [pdf]
Solutions
[pdf]
Week 3: Problem set 3
(17 Feb. 2012).
Questions [pdf]
Solutions
[pdf]
Week 4: Problem set 4
(24 Feb. 2012).
Questions [pdf]
Solutions
[pdf]
Week 5: Problem set 5
(2 Mar. 2012) FOR ASSESSMENT. Due on 9 March 2012 by 3pm.
Questions [pdf]
Week 6: Problem set 6
(9 Mar. 2012).
Questions [pdf]
Solutions
[pdf]
Week 7: Problem set 7
(16 Mar. 2012).
Questions [pdf]
Solutions
[pdf]
Week 8: Problem set 8
(23 Mar. 2012).
Questions [pdf]
Solutions
[pdf]
Week 9: Problem set 9
(20 Apr. 2012). NOTE: Part of the last two sections use the central limit theorem to be covered on Monday.
Questions [pdf]
Solutions
[pdf]
Week 10: Problem set 10
(27 Apr. 2012). NOTE: This is homework for ASSESSMENT. Deadline is May 4, by 4pm.
Questions [pdf]
Solutions
[pdf]
Week 11: Problem set 11
(4 May 2012).
Questions [pdf]
Solutions
[pdf]
Links
Some of you
may find the following link(s) useful. But I think that
textbooks are in general a better source of information. So
don't be lazy - go to the library! The internet is a great source of
information in terms of availability but it tends to be less so in
terms of quality.
Combinatorics
http://www.cse.unl.edu/~choueiry/S06-235/files/Combinatorics-Handout.pdf
http://www.maths.qmw.ac.uk/~pjc/notes/comb.pdf
Probability spaces
http://en.wikipedia.org/wiki/Probability_space
(Wikipedia)
http://www.ds.unifi.it/VL/VL_EN/prob/index.html
