To gain a solid knowledge of the basic aspects of probabilità theory: construction of probabilità measures on measurable spaces, 0-1 law, independence, conditional expectation, random variables, convergence of random variables, characteristic functions, central limit theorem, branching processes, discrete martingales.
teacher profile teaching materials
Introduction to measure theory: existence and uniqueness results for probability measures. First Borel–Cantelli lemma.
Random variables and their distriubutions.
Independence. Second Borel–Cantelli lemma. The 0-1 law.
Integration. Expected value. Monotone convergence theorem.
Dominated convergence theorem.
Jensen's inequality, Hoelder and Cauchy-Schwarz inequalities. Markov's inequality.
Examples of weak and strong laws of large numbers. Product spaces. Fubini's theorem. Joint laws.
Conditional expectation. Martingales and convergence theorems.
Discrete time stochastic processes. Gambilng. Optional stopping theorem and applications.
Convergence theorems for martingales. Examples and problems with martingales. Kolmogorov's strong law.
Convergence in distribution and characteristic functions. Central limit theorem. Modes of convergence of random variables.
R. Durrett, Probability: Theory and Examples. Thomson, (2000).
Fruizione: 20410414 CP410 - TEORIA DELLA PROBABILITÀ in Matematica LM-40 CAPUTO PIETRO, CANDELLERO ELISABETTA
Programme
An introductory example: the branching process.Introduction to measure theory: existence and uniqueness results for probability measures. First Borel–Cantelli lemma.
Random variables and their distriubutions.
Independence. Second Borel–Cantelli lemma. The 0-1 law.
Integration. Expected value. Monotone convergence theorem.
Dominated convergence theorem.
Jensen's inequality, Hoelder and Cauchy-Schwarz inequalities. Markov's inequality.
Examples of weak and strong laws of large numbers. Product spaces. Fubini's theorem. Joint laws.
Conditional expectation. Martingales and convergence theorems.
Discrete time stochastic processes. Gambilng. Optional stopping theorem and applications.
Convergence theorems for martingales. Examples and problems with martingales. Kolmogorov's strong law.
Convergence in distribution and characteristic functions. Central limit theorem. Modes of convergence of random variables.
Core Documentation
D. Williams, Probability with martingales. Cambridge University Press, (1991).R. Durrett, Probability: Theory and Examples. Thomson, (2000).
Type of delivery of the course
Lecture, exercise sessions, written solutions available online.Type of evaluation
2 intermediate written examinations (2 hours each) and 1 final written examination (3 hours)Fruizione: 20410414 CP410 - TEORIA DELLA PROBABILITÀ in Matematica LM-40 CAPUTO PIETRO, CANDELLERO ELISABETTA