The class aims at providing the basic knowledge concerning probability and it rules, discrete and continuous random variables (r.v.’s), and stochastic processes. The class contents are organized along these lines, addressing first probability basics, combinatorial analysis, discrete r.v.’s, continuous r.v.’s. (both along with the definition of 1st and 2nd moments and with examples of the main statistical distributions and probability density functions), multiple r.v.’s and their joint distribution, relevant inequalities and the Central Limit Theorem and, finally, the basics of Random Processes (stationarity, correlation and covariance functions).
The course introduces the key concepts related to stochastic modeling in the framework of disaster risk prevention and assessment. Basic knowledge will be provided about probability theory, random variables, stochastic processes, and Bayesian decision theory. Examples of applications to problems of data modeling and analysis associated with risk applications will be discussed.
AIMS AND LEARNING OUTCOMES
The main goal of the class is to provide basic knowledge on probability, random variables and random processes. Basic background knowledge on mathematical analysis is required for effectively following the course. More specifically, the lectures start with the basic definitions and concepts of probability, including conditional probability and independence, Bayes’ Rule, and some combinatorial analysis. The next topic is discrete r.v.’s, starting with the definition of Probability Mass Function and covering several specific cases (uniform, Bernoulli, Binomial, Poisson, …), along with the calculation of first and second moments. This is followed by the analogous treatment of continuous r.v.’s, with the definitions and cumulative distribution and probability density function (pdf), expectation and variance, and specific examples (uniform, exponential, normal,…). In both cases, multiple r.v.’s and their distributions are treated, as well. The last topics on r.v.’s include correlation and covariance, Markov and Chebyshev Inequalities and the Central Limit Theorem. The class concludes with the basics of Random Process, including the notion of stationarity, autocorrelation and covariance function. Several examples and problems are considered throughout. The list of the covered arguments is the following:
• Introduction to probability models
• Conditional probability and Bayes’ Rule
• Combinatorial analysis
• Discrete random variables
• Expectation and variance of discrete r.v.’s
• Multiple discrete r.v.’s
• Continuous r.v.’s
• Expectation and variance of continuous r.v.’s
• Multiple continuous r.v.’s
• Inequalities and Central Limit Theorem
• Random Processes
- Probability basics
- Basic definitions, probability rules, conditional probability, independence, combinatorial methods.
- Discrete random variables
- Probability Mass Function, expectation, variance; discrete distributions: uniform, Bernoulli, Binomial, Poisson; multiple discrete r.v.’s.
- Continuous random variables
- Cumulative distribution function, probability density function; expectation and variance; continuous distributions: uniform, exponential, normal; multiple continuous r.v.’s, joint distribution and density; functions of r.v.’s.
- Correlation, covariance; Markov and Chebyshev Inequalities; Central Limit Theorem.
- Random Processes.
- Random Processes and r.v.’s; mean value and variance; autocorrelation, autocovariance, correlation coefficient; complex processes; stationarity and properties of stationary processes.
- Course material on Aulaweb (https://www.aulaweb.unige.it): copy of all lecture slides
- Dimitri P. Bertsekas and John Tsitsiklis, Introduction to Probability, 2nd Ed., Athena Scientific,
Office hours: Appointment upon students' requests (direct or by email).
Office hours: Appointment upon students' requests.
RAFFAELE BOLLA (President)
FRANCO DAVOLI (President Substitute)
All class schedules are posted on the EasyAcademy portal.
The exam is mainly written (2-3 problems on the different topics treated); an oral exam can be agreed upon to improve (or not) the result.
Written examination and optionally oral examination.
|18/01/2022||10:00||SAVONA||Scritto + Orale|
|01/02/2022||10:00||SAVONA||Scritto + Orale|
|16/02/2022||10:00||SAVONA||Scritto + Orale|
|15/06/2022||10:00||SAVONA||Scritto + Orale|
|29/06/2022||10:00||SAVONA||Scritto + Orale|
|13/07/2022||10:00||SAVONA||Scritto + Orale|
|07/09/2022||10:00||SAVONA||Scritto + Orale|