Subject: STATISTICS AND ELEMENTS OF PROBABILITY (A.A. 2021/2022)
Unit Statistica ed elementi di probabilità
This course aims at providing basic mathematical knowledge useful for analyzing problems, building models, working out and assessing solutions, when dealing with random phenomena.
For a fuller understanding of the training objectives, see the expected learning outcomes.
A basic knowledge of the theory of numerical series and of integro-differential calculus for real functions of one real variable, as provided by the propaedeutic Mathematical Analysis class.
The contents are structured in four macro-topics as described by the following list:
1) ELEMENTARY PROBABILITY AND RANDOM VARIABLES (3CFU)
Introduction. Sample spaces and events. The family of events. The meaning of probability. The axioms of probability. Equiprobable spaces. Binomial coefficients. Diffuse probabilities. Conditional probability. Composite and total probabilities. Bayes formula. Independence of two events. Independence of three or more events. Distribution functions. Quantile functions. Discrete random variables. Continuous random variables. Transformations of random variables. Mean value. Mean, median and mode. Variance and standard deviation. Markov and Chebyshev inequalities. Moments and their generating function.
2) SPECIAL DISTRIBUTIONS (1CFU)
Finite Bernoulli trials. Infinite Bernoulli trials. Dichotomous urns. Poisson distributions and processes. Exponential and gamma distributions. The uniform model. The Gaussian model. Complements on the Gaussian model.
3) RANDOM VECTORS AND LIMIT THEOREMS (1CFU)
Bidimensional random vectors. Discrete vectors and independence. Means of sums and products. Covariance and correlation. Random samples. Sample distributions. Large samples. Convergence in law.
4) PARAMETER ESTIMATION (1CFU)
Statistical inference. Point estimation. The method of moments. Estimation of a mean. Estimation of a standard deviation. Estimation of an intensity. Estimation of a proportion. Estimation of a supremum.
The breakdown of CFU by macro-topic and the list of topics in each macro-topic are to be understood as indicative: they may be subject to changes during the lessons, based on contingencies and the feedback received.
For a fuller understanding of the contents, see the reference texts.
Teaching is delivered, face to face, in Italian. Teaching methods include: classroom lectures open to discussion; exercise solving assignments; office hours. Attendance is not compulsory.
The exam will take place at the end of the lessons according to the official exam schedule. It will consist of a single oral test (with the aid of pen and paper or other writing instruments) lasting approximately one hour and structured in two stages. In the first stage the student will have to solve one or more exercises, proposed by the teacher on the contents of the first two macro-topics, demonstrating to be able to reach a solution with autonomy of judgment and mastery of the methods and models applied. In the second stage the student will have to discuss the contents of the last two macro-topics, demonstrating that they have investigated them and that they are able to communicate them effectively; the discussion will start with a topic chosen by the student and it will continue according to the teacher's instructions. The grade will be based on an overall assessment of what the student has demonstrated. During the test students will be allowed to consult their favorite reference material, but the test will be strictly individual.
Knowledge and understanding: students will have knowledge of the basic mathematical methods and models for the analysis of random phenomena and they will have understood their main properties.
Applying knowledge and understanding: students will be able to analyze concrete problems and describe them in mathematical form, so as to solve them with appropriate calculations.
Making judgements: students will be able to source information for formulating and solving problems; they will be able to critically assess the results achieved, also based on the choices they made.
Communication: students will be able to present, clearly and synthetically, the results of their work.
Lifelong learning skills: students will have learned to investigate topics in probability and statistics.
Boella, M. (2020). Probabilità e Statistica per Ingegneria e Scienze. Pearson Italia, Milano-Torino.
Disponibile presso la Biblioteca Scientifica Interdipartimentale (SALA MATEM A.17/873).
Diapositive a cura del docente.
Disponibili su Dolly (con una breve descrizione dei contenuti e l'indicazione delle letture suggerite).
Tavole delle distribuzioni ed esercizi a cura del docente.
Disponibili su Dolly.
Il materiale su Dolly sarà reso disponibile prima dell'inizio delle lezioni, ma sarà da considerare in forma definitiva solo al termine delle stesse.