- Uniform probabilities
- Finite and discrete probability spaces
- Independence and conditional probability
- Existence of probabilities
- Random variables: independence, mean and standard variation
- Joint distributions, covariance and correlation
- Markov chains
- Probabilities in the continuum: properties, random variables, moments, joint distributions
- Inequalities and limit theorems
- Kolmogorov axiomatic definition of probability
- Other useful references:
Ronald Meester - A Natural Introduction to ProbabilityTheory, Wiley 2003
Venkatesh,The theory of probability explorations and applications, Cambridge University Press (2012)
F. Caravenna, P. Dai Pra. Probabilita’. Un'introduzione attraverso modelli e applicazioni. Springer-Verlag Italia (2013)
Learning Objectives
Knowledge acquired: the successful student has a basic knowledge of elementary probability theory and of several examples and applications. He/she has faced several questions about the foundation of probability theory and on the first difficulties in using the theory.
Competence acquired: the successful student is able to rigorously solve several elementary problems about determining discrete and continuous probabilities.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required: Analisi Matematica II (Mathematical Analysis II - multivariate calculus)
Courses recommended: Analisi Matematica III (Mathematical analysis III - Lebesgue integration)
Teaching Methods
CFU: 6
Number of hours for personal study and other individual learning: 100
Number of hours for classroom activities: 52
Number of hours for laboratory activities (laboratory classes): 0
Number of hours for topics other exercises (laboratory and field): 0
Number of hours for seminars to: 0
Number of hours related to work experience: 0
Number of hours per course tests: 0
Further information
Frequency of lessons and exercises: Not required
Tools for Teaching:
http://web.math.unifi.it/users/gandolfi/didindex.html
1) Uniform probabilities
- Definition of probability
- Combinatorics
(2) Finite and discrete probability spaces
- Generalization of probability
- Inclusion-exclusion formula
(3) Independence and conditional probability
- Independence of two events
- Collective independence
- Conditional probability
- Bayes formula
- Conditional independence
(4) Existence of probabilities