Elementary probability theory: discrete distributions, repeated trials, continuous random variables. Some basic limit theorems and introduction to Markov chains.
Curriculum
teacher profile teaching materials
2. Introduction to Probability.
3. Conditional probability, Bayes' formula. Independence.
4. Discrete random variables. Bernoulli, binomials, Poisson, geometric, hipergeometric, negative binomial.
Expected value.
5. Continuous random varaibles. Uniform, exponential, gamma, gaussian.
Expected value.
6. Independent variables and joint laws.
Sum of two or more independent random variables. Poisson process. Maxima and minima of independent random variables.
7. Limit theorems. Markov and Chebyshev inequalities.Weak law of large numbers. Generating functions and a sketch of proof of the central limit theorem.
F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).
Programme
1. Introduction to combinatorial analysis.2. Introduction to Probability.
3. Conditional probability, Bayes' formula. Independence.
4. Discrete random variables. Bernoulli, binomials, Poisson, geometric, hipergeometric, negative binomial.
Expected value.
5. Continuous random varaibles. Uniform, exponential, gamma, gaussian.
Expected value.
6. Independent variables and joint laws.
Sum of two or more independent random variables. Poisson process. Maxima and minima of independent random variables.
7. Limit theorems. Markov and Chebyshev inequalities.Weak law of large numbers. Generating functions and a sketch of proof of the central limit theorem.
Core Documentation
Sheldon M. Ross, Calcolo delle Probabilita'. Apogeo, (2007).F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).
Reference Bibliography
Sheldon M. Ross, Calcolo delle Probabilita'. Apogeo, (2007). F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).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 written final examination (3 hours). teacher profile teaching materials
F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Programme
Refer to the web-page of the courseCore Documentation
Sheldon M. Ross, Calcolo delle probabilita' (Apogeo Ed.)F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Reference Bibliography
Sheldon M. Ross, Calcolo delle probabilita' (Apogeo Ed.) F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)Type of delivery of the course
Exercises are solved on the blackboard - the proposed exercises aim at clarifying and practicing with the concepts seen during the lecturesType of evaluation
Refer to the web-page of the course teacher profile teaching materials
2. Introduction to Probability.
3. Conditional probability, Bayes' formula. Independence.
4. Discrete random variables. Bernoulli, binomials, Poisson, geometric, hipergeometric, negative binomial.
Expected value.
5. Continuous random varaibles. Uniform, exponential, gamma, gaussian.
Expected value.
6. Independent variables and joint laws.
Sum of two or more independent random variables. Poisson process. Maxima and minima of independent random variables.
7. Limit theorems. Markov and Chebyshev inequalities.Weak law of large numbers. Generating functions and a sketch of proof of the central limit theorem.
F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).
Programme
1. Introduction to combinatorial analysis.2. Introduction to Probability.
3. Conditional probability, Bayes' formula. Independence.
4. Discrete random variables. Bernoulli, binomials, Poisson, geometric, hipergeometric, negative binomial.
Expected value.
5. Continuous random varaibles. Uniform, exponential, gamma, gaussian.
Expected value.
6. Independent variables and joint laws.
Sum of two or more independent random variables. Poisson process. Maxima and minima of independent random variables.
7. Limit theorems. Markov and Chebyshev inequalities.Weak law of large numbers. Generating functions and a sketch of proof of the central limit theorem.
Core Documentation
Sheldon M. Ross, Calcolo delle Probabilita'. Apogeo, (2007).F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).
Reference Bibliography
Sheldon M. Ross, Calcolo delle Probabilita'. Apogeo, (2007). F. Caravenna, P. Dai Pra, Probabilita'. Springer, (2013).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 written final examination (3 hours). teacher profile teaching materials
F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Programme
Refer to the web-page of the courseCore Documentation
Sheldon M. Ross, Calcolo delle probabilita' (Apogeo Ed.)F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Reference Bibliography
Sheldon M. Ross, Calcolo delle probabilita' (Apogeo Ed.) F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)Type of delivery of the course
Exercises are solved on the blackboard - the proposed exercises aim at clarifying and practicing with the concepts seen during the lecturesType of evaluation
Refer to the web-page of the course