Menù principale
B018779 - PROBABILITY
Main information
Teaching Language
Course Content
Suggested readings
Learning Objectives
Prerequisites
Teaching Methods
Type of Assessment
Course program
Academic Year 2023-24
Coorte 2023 - Second Cycle Degree in MATHEMATICS
Course year
First year - Second Semester
Belonging Department
Mathematics and Computer Science "Ulisse Dini" (DIMAI)
Course Type
Single education field course
Scientific Area
MAT/06 - PROBABILITY AND STATISTICS
Credits
9
Teaching Hours
72
Teaching Term
19/02/2024 ⇒ 14/06/2024
Attendance required
No
Type of Evaluation
Final Grade
Course Content
show
Course program
show
Lectureship
Teaching Language
Italian, and there is the possibility to request the course and the material in English.
Course Content
Law of large numbers, Central limit theorem.Large deviation theory. Rigorous probability theory. Weak convergence. Caratteristic function, Decomposition of probability laws. Conditional Probability. Martingales. Stochastic processes: Poisson, gambling games, Markov chains, Random walk, Branching processes, Brownian motion.
Suggested readings (Search our library's catalogue)
1) Rosenthal, A first look at rigorous probability theory.
2) Frank den Hollander, Large deviations, Chapters 1 e 2.
3) Baldi, Calcolo delle Probabilità. Capitolo 5.
4) Caravenna Dai Pra, Probabilita`.
5) Sheldon Ross, Calcolo delle probabilità.
6) van der Hofstad, random graphs
2) Frank den Hollander, Large deviations, Chapters 1 e 2.
3) Baldi, Calcolo delle Probabilità. Capitolo 5.
4) Caravenna Dai Pra, Probabilita`.
5) Sheldon Ross, Calcolo delle probabilità.
6) van der Hofstad, random graphs
Learning Objectives
The course aims to provide the students with fundamental knowledge and understanding about rigorous probability theory for general random variables and stochastic processes. One of the aim is to let the students develop technical skills to apply the knowledge and results to model concrete situations that needs general random variables and to compute probabilities and distributions requested in the situations described in the problem.
The course aims to provide the students with fundamental knowledge and understanding about limit theorems such as Strong Law of large numbers, central limit Theorems and large deviation theorems (with their proofs) with particular emphasis to the different hypothesis one could assume and to the different type convergence one could obtain. One of the aim is to let the students develop technical skills needed to perform asymptotic estimates, to solve concrete problems and to estimate probabilities that cannot be exactly computed. Another aim is to let the students develop technical skills needed to compute quantities related to Poisson processes and Markov chains and provides methods and examples for the solution of exercises that require to model concrete situation with those stochastic processes. Special attention will be paid to help the students to develop communication skills needed to explain with rigorous mathematical language the results explained and proved in class and to give justification of the methods used to solve the exercises. The course aims to stimulate students to develop independent and critical thinking to establish the appropriate results that can be used depending on the concrete situation described in the problem.
The course aims to provide the students with fundamental knowledge and understanding about limit theorems such as Strong Law of large numbers, central limit Theorems and large deviation theorems (with their proofs) with particular emphasis to the different hypothesis one could assume and to the different type convergence one could obtain. One of the aim is to let the students develop technical skills needed to perform asymptotic estimates, to solve concrete problems and to estimate probabilities that cannot be exactly computed. Another aim is to let the students develop technical skills needed to compute quantities related to Poisson processes and Markov chains and provides methods and examples for the solution of exercises that require to model concrete situation with those stochastic processes. Special attention will be paid to help the students to develop communication skills needed to explain with rigorous mathematical language the results explained and proved in class and to give justification of the methods used to solve the exercises. The course aims to stimulate students to develop independent and critical thinking to establish the appropriate results that can be used depending on the concrete situation described in the problem.
Prerequisites
Differential and integral calculation in one variable for real functions. Basic knowledge of algebra and geometry.
Basic notions of discrete and continuous random variables.
Basic notions of discrete and continuous random variables.
Teaching Methods
Lectures and discussion and correction of homework.
Type of Assessment
The exam consists of a written and oral examinations per session. The written exam will have open questions of two types. The first type in which the student should state and prove results explained during the lessons, with the aim to verifying the knowledge, the understanding and the quality of the exposition. A second type in which the questions are conceived to assess the ability of the students to apply their skills to problem modelling and solving, and to give the rigorous justification using formule and the appropriate scientific language.
The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. During the esposition of the definitions and results with their proofs the student has to show the degree of comprehension of the theoretical and applied aspects of the topics treated during the course. In the assessment, special attention is paid to communication skills, appropriate use of mathematical language. In particular, the teacher will pose be short questions on the relations between topics and on possible strategies to adapt the proofs to different hypothesis in order to evaluate the autonomy and critical thinking on the course topics.
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.
The oral examination is designed to evaluate the degree of understanding of the theory presented in the course. During the esposition of the definitions and results with their proofs the student has to show the degree of comprehension of the theoretical and applied aspects of the topics treated during the course. In the assessment, special attention is paid to communication skills, appropriate use of mathematical language. In particular, the teacher will pose be short questions on the relations between topics and on possible strategies to adapt the proofs to different hypothesis in order to evaluate the autonomy and critical thinking on the course topics.
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
1) Limit theorems: Strong and weak law of large numbers, Central limit theorem. The normal approximation.
-Books:
Rosenthal, A first look at rigorous probability theory.
2) Large deviation theory for sequences of independent and identically distributed random variables.
a) Large deviation theory of the empirical average S_n/n, Cramer theorem.
b) Large deviation theory of the empirical measure, Sanov theorem.
-Book:
Large deviations, Frank den Hollander Chapter 1 and 2.
3) Probability theory using measure theory
-Book:
1) Rosenthal, A first look at rigorous probability theory.
Chapter 1: The need for measure theory. The uniform distribution and non-measurable sets.
Chapter 2 Probability triples. Basic definition. Constructing probability triples.The Extension Theorem. Constructing the Uniform[0,1] distribution. Extensions of the Extension Theorem. Coin tossing and other measures.
Chapter 3: Random variables, Independence, Continuity of probabilities. Limit events. Tail fields and Kolmogorov theorem.
Chapter 4: Expected values: simple random variables, General non-negative random variables, arbitrary random variables. The integration connection.
Chapter 5: Inequalities and convergence. Various inequalities. Convergence of random variables, Weak Laws of large numbers. Eliminating the moment conditions.
Chapter 6: Distributions of random variables. Change of variable theorem. Examples of distributions..
Chapter 9: More probability theorems. Limit theorems. Differentiation of expectation. Moment generating functions. Fubini's Theorem and convolution.
4) Stochastic processes: Poisson process. Gambling games. Markov chains, Random walks, Branching, Moto browniano.
4a) Processo di Poisson definizione e proprieta` collegamenti variabili aleatorie studiate in precedenza.
-Book: Calcolo delle probabilita` Ross,
Chapter 9: Poisson Process: Definition, Erlang-Poisson formula and theorem.
4b) Stochastic processes and gambling games.
-Book Rosenthal
Chapter 7: A first existence theorem, Gambling and gambler's ruin, Gambling policies..
4c) Markov chains and Random walks
Books
-Rosenthal
Chapter 8: Definition of discrete Markov chains and Examples. A Markov chain existence theorem. Random walk. Transience, recurrence, and irreducibility. Stationary distributions and convergence. Existence of stationary distributions examples and exercises.
-Baldi: chapter 5
Finite state space Markov chains. Examples and exercises.
4d) Branching processes,
-Book vd Hofstad Chapter 3.
4e) Processi processes and Brownian motion.
-Book Rosenthal
Chapter 15.
5) Weak convergence. Characteristic functions. Decomposition of probability laws. Conditional probability and expectation. Martingales.
-Book Rosenthal
Chapter 10: Definition of weak convergence and Equivalences of weak convergence. Connections to other convergence.
Chapter 11: Characteristic functions. The continuity theorem. The Central Limit Theorem. Generalisations of the Central Limit Theorem and Method of moments.
Chapter 12: Decomposition of probability laws.Lebesgue and Hahn decompositions. Decomposition with general measures.
Chapter 13: Conditional probability and expectation. Conditioning on a random variable. Conditioning on a sub-er-algebra. Conditional variance.
Chapter 14: Martingales. Stopping times. Martingale convergence. Maximal inequality and Teorema di Wald.
-Books:
Rosenthal, A first look at rigorous probability theory.
2) Large deviation theory for sequences of independent and identically distributed random variables.
a) Large deviation theory of the empirical average S_n/n, Cramer theorem.
b) Large deviation theory of the empirical measure, Sanov theorem.
-Book:
Large deviations, Frank den Hollander Chapter 1 and 2.
3) Probability theory using measure theory
-Book:
1) Rosenthal, A first look at rigorous probability theory.
Chapter 1: The need for measure theory. The uniform distribution and non-measurable sets.
Chapter 2 Probability triples. Basic definition. Constructing probability triples.The Extension Theorem. Constructing the Uniform[0,1] distribution. Extensions of the Extension Theorem. Coin tossing and other measures.
Chapter 3: Random variables, Independence, Continuity of probabilities. Limit events. Tail fields and Kolmogorov theorem.
Chapter 4: Expected values: simple random variables, General non-negative random variables, arbitrary random variables. The integration connection.
Chapter 5: Inequalities and convergence. Various inequalities. Convergence of random variables, Weak Laws of large numbers. Eliminating the moment conditions.
Chapter 6: Distributions of random variables. Change of variable theorem. Examples of distributions..
Chapter 9: More probability theorems. Limit theorems. Differentiation of expectation. Moment generating functions. Fubini's Theorem and convolution.
4) Stochastic processes: Poisson process. Gambling games. Markov chains, Random walks, Branching, Moto browniano.
4a) Processo di Poisson definizione e proprieta` collegamenti variabili aleatorie studiate in precedenza.
-Book: Calcolo delle probabilita` Ross,
Chapter 9: Poisson Process: Definition, Erlang-Poisson formula and theorem.
4b) Stochastic processes and gambling games.
-Book Rosenthal
Chapter 7: A first existence theorem, Gambling and gambler's ruin, Gambling policies..
4c) Markov chains and Random walks
Books
-Rosenthal
Chapter 8: Definition of discrete Markov chains and Examples. A Markov chain existence theorem. Random walk. Transience, recurrence, and irreducibility. Stationary distributions and convergence. Existence of stationary distributions examples and exercises.
-Baldi: chapter 5
Finite state space Markov chains. Examples and exercises.
4d) Branching processes,
-Book vd Hofstad Chapter 3.
4e) Processi processes and Brownian motion.
-Book Rosenthal
Chapter 15.
5) Weak convergence. Characteristic functions. Decomposition of probability laws. Conditional probability and expectation. Martingales.
-Book Rosenthal
Chapter 10: Definition of weak convergence and Equivalences of weak convergence. Connections to other convergence.
Chapter 11: Characteristic functions. The continuity theorem. The Central Limit Theorem. Generalisations of the Central Limit Theorem and Method of moments.
Chapter 12: Decomposition of probability laws.Lebesgue and Hahn decompositions. Decomposition with general measures.
Chapter 13: Conditional probability and expectation. Conditioning on a random variable. Conditioning on a sub-er-algebra. Conditional variance.
Chapter 14: Martingales. Stopping times. Martingale convergence. Maximal inequality and Teorema di Wald.