Probability Theory

Module Title: Introduction to Probability and Information
Objectives: to present the basic concepts of probability theory and
information theory in an integrated format, and to provide students
with the background to pursue independent research in these
Target audience: first and second year PhD students in
Engineering and the Mathematical Sciences.
Teaching methods: 24 hours of lectures, together with guided
study assignments
Prerequisites: undergraduate multi-variable calculus and linear
algebra, some knowledge of real analysis and basic probability
Module content:
1) Elementary probability, including events, conditional probability,
random variables, expectation
2) Markov chains
3) Applications
4) Probability triples, including sigma algebras
5) Limit theorems, including modes of convergence, Borel-Cantelli
Lemmas and 0-1 Law, law of large numbers, central limit theorem
6) Entropy and mutual information
7) Asymptotic equipartition property and typical sequences
8) Data compression and applications
9) Channel capacity
10) Gaussian channels
11) Sanov's theorem and applications in statistics
12) Kolmogorov complexity
Assessment: one 2-hour examination
P. Billingsley,
“Probability and Measure'', third edition. Wiley (1995).
[Wiley series in probability and mathematical statistics]
J. S. Rosenthal,
“A first look at rigorous probability theory'',
second edition. World Scientific (2006).
Y. Suhov and M. Kelbert,
“Probability and Statistics by Example'', Volume 1, Basic Probability
and Statistics Cambridge University Press (2005)
T. M. Cover and J. A. Thomas,
“Elements of Information Theory'',
second edition. Wiley (2006).
A. Y. Khinchin,
“Mathematical Foundations of Information Theory'',
Dover, New York (1957).