Learning objectives
At the end of the course the student should be able to manage the basic concepts of Probability, both in a theoretical frame and in applications. Those aspects of the Physics in the first three-years training which, explicitly or implicitly, touch probabilistic themes should be faced in this course. Moreover, a survey on possible developments in advanced courses or circumstances will also be presented.
Prerequisites
Methematical courses of the previous two years.
Course unit content
The course will present ab initio the Theory of Probability, giving the basic mathematical tools and the conceptual items necessary to the students. Application to Physics will deserve a particular attention, but also a more general kind of problems in Statistics and Information Theory will be considered. The analytical treatment of various arguments will be enriched by exercises of numerical simulations.
Full programme
Some historical considerations on Probability.
Probability Spaces, recalls on measurable algebras and theory of measure.
Elementary outcomes, conditional probability, independence.
Bayes formula.
Combinatorics, binomial distribution, Bernoulli processes.
Other discrete distributions.
Discrete random variables, expectation and variance.
Density, repartition function. Independence and covariance.
Functions of random variables. Continuous random variables.
Chebyshev inequality.
Exponential, uniform and normal distributions. Poisson processes. Analysis of sequences generated along various statistical laws.
Sampling. Confidence interval.
Law of large numbers, Central Limit Theorem.
Markov chains and processes. Classification of states. Ergodic chains.
Information and entropy, partitions formalism.
Probability and chaos.
Application to discrete processes. Probabilistic automata.
Analytical and numerical methods in treating Time Series.
An introduction to continuous random processes.
Reading of a paper on the subjects of the course.
Bibliography
G.Boffetta, A. Vulpiani
Probabilità in Fisica
Springer, 2012.
K. Baclawski and G-C Rota
An Introduction to Probability and Random Processes.
recover on www.freescience.info, or
http://www.ellerman.org/gian-carlo-rota/
G.Cicuta
Notes from lessons in previous courses
E. Onofri
Metodi Probabilstici della Fisica
recover on http://www.freescience.info
Teaching methods
Therory and exercises will be presented at the same time during lessons. Some exercises, to be completed as a homework, will be discussed all together in the classroom.
Assessment methods and criteria
The final examination will consist in written exercises on the same arguments presented in the course.
Those students which did not actively participate to the course shall undergo a longer examination, in order to prove their knowledge.
This way, kwowledge and understanding will be verified, also in view of applications to typical problems of the field. Skill in communication will also be checked.
Other information
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