32827 - Theory of Probability

Learning outcomes

At the end of the course, the candidate will be able to apply the basic tools of probability, especially the most useful within statistical analysis. Specifically, he will be able to calculate the probability of complex events, to manage a wide range of discrete and continuous random variables, and to know and apply the main discrete random process patterns.

Course contents

Random trials, events and logical operations. Combinatorics. Axioms and elementary probability. Conditional probability, independence and Bayes theorem. Discrete random variables and related models (Binomial, Geometric, Hypergeometric, Pascal, Poisson). Continuous random variables and related models (Uniform, Exponential, Gamma and Beta). Gaussian variables and derived distributions (Log-normal, Chi squared, Student t-distribution, Snedecor-Fisher F-distribution). Pareto Distribution. Bivariate discrete random variables. Covariance and its properties. Sequences and convergences of random variables. Bernoulli Theorem and Central Limit Theorem. Graduation function. Ordinal variables. Moment-generating function. Discrete and continuous Random Processes. Random walks. Poisson Processes with applications. Markov chains and classification of states. Definition of martingale process and brownian motion.

Readings/Bibliography

- Maurizio Brizzi. Calcolo delle probabilità con note introduttive di inferenza statistica. Editrice Lo Scarabeo, Bologna, 2004 (only in Italian).

- Maurizio Brizzi. Introduzione al calcolo delle probabilità e all'inferenza statistica. Libreriauniversitaria.it, Limena (PD), 2014 (only in Italian).

- Geoffrey Grimmett and David Stirzaker. Probability and Random processes. Oxford University Press, 2001.

Teaching methods

Direct teaching and at least 4-6 hours of laboratory work.

Assessment methods

Written test including three exercises with numerical applications. Each exercise can be valued between 9 and 12 points. Usually the first exercise is related to elementary probability, the second involves random variables, the third concerns stochastic processes. Oral test is dedicated to the complete course theory, jointly with some quick numerical examples.

- Evaluation scale:

30 e lode (A+) =excellent

29 - 30 (A) = very good

27 - 28 (B+) = good

25 - 26 (B) = fairly good

22 - 24 (C) = more than sufficient

20 - 21 (D) = sufficient

18 - 19 (E) = barely sufficient

Teaching tools

Working sheets in Power Point are available to students, covering all the Course topics and containing theoretical features, examples and exercises (even in English, if requested).

Links to further information

https://www.unibo.it/sitoweb/maurizio.brizzi/

Office hours

See the website of Maurizio Brizzi