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.
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.
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
CFU: 6
Number of hours for personal study and other individual learning: 100
Number of hours for classroom activities: 52
Number of hours for laboratory activities (laboratory classes): 0
Number of hours for topics other exercises (laboratory and field): 0
Number of hours for seminars to: 0
Number of hours related to work experience: 0
Number of hours per course tests: 0
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.
Suggested readings
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