Combinatorics. Elementary probability on finite sets. Probability measures: events, sigma-algebras, measures. Conditional probability.
Random variables: distribution and distribution function. Expected value, variance, Markov and Chebyshev inequalities.
Discrete and absolutely continuous distributions.
Vector valued random variables. Covariance and correlation coefficient. Stochastic independence. Laws of large
numbers. Conditional expectation.
Giuseppe Modica, Laura Poggiolini Note di Calcolo delle Probabilità. Pitagora Editrice.
Ambrosio - Da Prato - Mennucci, SNS Lecture Notes
Sempi - Introduzione alla Probabilità
Sempi - Secondo Corso di Probabilità
Learning Objectives
Knowledge acquired: the successful student has a basic knowledge of elementary probability theory and of several examples and applications. He/she has faced several questions about the foundation of probability theory and on the first difficulties in using the theory.
Competence acquired: the successful student is able to rigorously solve several elementary problems about determining discrete and continuous probabilities.
Prerequisites
Courses to be used as requirements (required and/or recommended)
Courses required: Analisi Matematica II (Mathematical Analysis II - multivariate calculus)
Courses recommended: Analisi Matematica III (Mathematical analysis III - Lebesgue integration)
Teaching Methods
Front teaching
Type of Assessment
Written and Oral examination.
The written part of the final exam can be substituted by written proofs held during the course
Course program
Combinatorics: permutations, lists and functions, drawings and grouping.
Elementary probability on finite sets.
Probability measures: events, sigma-algebras, measures. Probabilities on finite sets, on denumerable sets. Uniform probability on intervals.
Conditional probability: Bayes formula and total probabilities formula.
Random variables: definition, distribution and distribution function. Typical classes,
Integral and distributions. Expected value. Integral of compositions of random variables. Cavalieri formula. Variance, Markov and Chebyshev inequalities.
Examples of discrete distributions: Dirac delta, Bernoulli distribution, binomial distribution, hypergeometric distribution. Negative binomial distribution. Poisson distribution and rare events. Geometric and modified geometric distribution. Memorylessness.
Examples of absolutely continuous distributions: uniform distribution. Normal distribution. Exponential distribution and memorylessness. Gamma distributions.
Vector valued random variables: joint distribution and marginal distributions. Composition. Covariance and correlation coefficient. Stochastic independence: independent
events and independent random variables. Distribution of the sum of independent random variables.
Sequences of random variables: different notions of convergence. Weak law of large
numbers. Borel-Cantelli lemma and strong law of large numbers. Montecarlo method.
If time permits: Conditional expectation. Characteristic function and central limit
theorem. An introduction to Markov chains.