Probability space. Discrete and continuous models: random variables, their distributions, cumulative distribution functions, joint distributions and independence. Determination of density functions, moments, probability generating functions and characteristic functions. Convergence and approximation: Strong law of large numbers, Central limit theorem, applications to statistics. Markov chains.
Adopted:
-Baldi, Calcolo delle probabilita`
Suggested:
-S. Ross, Calcolo delle probabilita`
-Caravenna e Dai Pra, Probabilità
Learning Objectives
The course aims to provide the students with fundamental knowledge and understanding about discrete and continuous models and
random variables, in order to compiute probabilities needed in concrete situations and to compiute probability laws of one dimensional and multidimensional random variables in the case of known models and in the case in which there situation cannot be described via a known model. The course aims to provide the students with fundamental knowledge and understanding of limit theorems such as the Central limit theorem and the strong law of large numbers and their proofs. One of the goals is to let the students develop basic technical skills, and critical thinking, needed when modeling concrete situations and solving non deterministic mathematical problems for which it is needed to estimate the required probabilities that cannot be exactly computed.
The course aims to provide the students with fundamental knowledge and understanding about two basic examples of stocastic processes: Poisson process and Markov chains that are useful for Monte Carlo method. Another aim is to let the students develop basic technical skills, and critical thinking, needed when modeling concrete situations and solving non deterministic problems involving stochastic processes. Special attention will be paid to help the students to develop communication skills necessary to rigorously present the acquired knowledge in mathematical language and to jiustify clearly and precisely the solved exercises.
Prerequisites
Differential and integral calculation in one variable for real functions. Basic knowledge of algebra and geometry.
Teaching Methods
Lectures and discussion and correction of homework
Type of Assessment
The exam consists of a written and oral examinations per session. In the written exam will have open questions in which the student has to show his knowledge and his ability to apply the results to solve problems and give their justification using formule and the appropriate scientific language. Moreover, the questions will be formulated to highlight whether the student is able to choose the best probabilistic models to solve the concrete situation described in the exercise.
During the oral examination is designed to evaluate the degree of understanding of the theory presented in the course. In particular with the esposition of the definitions and results the teacher will verify the degree of comprehension of the theoretical and applied aspects. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Additionally, the students will have the opportunity to perform two partial tests that, if both successful, will allow them to access directly to the oral examination.
Course program
-Spazi di probabilita`: Definizione di spazi di probabilita`, anche nel caso uniforme, proprieta` generali, probabilita` condizionale, indipendenza, calcolo comninatorio.
-Modelli discreti: variabili aleatorie discrete e loro distribuzioni, funzioni di ripartizione, leggi congiunte e indipendenza, calcoli con densita`, speranza matematica, momenti, varianza e covarianza, legge dei grandi numeri, funzioni generatrici delle probabilita`.
-Modelli continui: variabili aleatorie continue, loro densita` e loro funzione di ripartizione con proprieta`, calcolo di leggi, densita` congiunte e calcolo con densita` congiunte, speranza matematica, momenti. Leggi normali, Gamma. Tempi di attesa e processo di Poisson. Generatori aleatori, simulazione. Speranza condizionale, funzione generatrici di momenti e trasformata di Laplace. Funzioni caratteristiche e Leggi normali multivariate.
-Convergenza e approssimazione. Legge dei grandi numeri, convegenza in legge e teorema del limite centrale, cenni alla statistica e ai problemi di stima.
-Catene di Markov, definizioni e generalita`, calcolo di leggi congiunte, classificazione degli stati, probabilita` invarianti, L'algoritmo di Metropolis, simulated annealing, problemi di passaggio.