Elementary probability theory: discrete distributions, repeated trials, continuous random variables. Some basic limit theorems and introduction to Markov chains.
Curriculum
teacher profile teaching materials
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Programme
Combinatorics, assioms of probability, conditional probability and independence, discrete random variables. Continuous random variables, density and distribution functions. Independence and joint laws. Relation between exponential distribution and Poisson distribution. Limit theorems, moment generating functions and sketch of central limit theorem.Core Documentation
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Reference Bibliography
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.) - F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)Type of delivery of the course
Preferably in presenceAttendance
Preferably in presenceType of evaluation
The written part mainly consists of exercises, but there can be questions about the theoretical parts seen in class (check with the lecturer for further details). teacher profile teaching materials
combinations, examples.
Probability Axioms. Sample spaces, events, probability axioms. Equally likely events and other examples.
Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
Discrete Random Variables. Bernoulli, binomial, and Poisson random variables.
Poisson process. Other discrete distributions: geometric, hypergeometric, negative binomial. Expected value and variance of a discrete random variable. Examples.
Continuous Random Variables. Probability density function and distribution function. Uniform distribution on an interval, exponential, gamma, normal, Weibull, Cauchy distributions. Link between gamma distributions and Poisson process. Expected value and variance for continuous random variables.
Joint Distributions and Independent Random Variables. Joint distributions, independent random variables. Density of the sum of two independent random variables. Convolution product for normal, gamma, Poisson distributions. Maximum and minimum of independent random variables.
Limit Theorems. Markov's and Chebyshev's inequalities. Weak law of large numbers. Moment generating function and a brief proof of the Central Limit Theorem.
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
- W. Feller, An introduction to probability theory and its applications (Wiley, 1968).
Programme
Combinatorial Analysis. Introduction to combinatorial calculations: permutations,combinations, examples.
Probability Axioms. Sample spaces, events, probability axioms. Equally likely events and other examples.
Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
Discrete Random Variables. Bernoulli, binomial, and Poisson random variables.
Poisson process. Other discrete distributions: geometric, hypergeometric, negative binomial. Expected value and variance of a discrete random variable. Examples.
Continuous Random Variables. Probability density function and distribution function. Uniform distribution on an interval, exponential, gamma, normal, Weibull, Cauchy distributions. Link between gamma distributions and Poisson process. Expected value and variance for continuous random variables.
Joint Distributions and Independent Random Variables. Joint distributions, independent random variables. Density of the sum of two independent random variables. Convolution product for normal, gamma, Poisson distributions. Maximum and minimum of independent random variables.
Limit Theorems. Markov's and Chebyshev's inequalities. Weak law of large numbers. Moment generating function and a brief proof of the Central Limit Theorem.
Core Documentation
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
- W. Feller, An introduction to probability theory and its applications (Wiley, 1968).
Type of delivery of the course
blackboard lecturesType of evaluation
final written exam and partial examinations teacher profile teaching materials
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Programme
Combinatorics, assioms of probability, conditional probability and independence, discrete random variables. Continuous random variables, density and distribution functions. Independence and joint laws. Relation between exponential distribution and Poisson distribution. Limit theorems, moment generating functions and sketch of central limit theorem.Core Documentation
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
Reference Bibliography
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.) - F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)Type of delivery of the course
Preferably in presenceAttendance
Preferably in presenceType of evaluation
The written part mainly consists of exercises, but there can be questions about the theoretical parts seen in class (check with the lecturer for further details). teacher profile teaching materials
combinations, examples.
Probability Axioms. Sample spaces, events, probability axioms. Equally likely events and other examples.
Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
Discrete Random Variables. Bernoulli, binomial, and Poisson random variables.
Poisson process. Other discrete distributions: geometric, hypergeometric, negative binomial. Expected value and variance of a discrete random variable. Examples.
Continuous Random Variables. Probability density function and distribution function. Uniform distribution on an interval, exponential, gamma, normal, Weibull, Cauchy distributions. Link between gamma distributions and Poisson process. Expected value and variance for continuous random variables.
Joint Distributions and Independent Random Variables. Joint distributions, independent random variables. Density of the sum of two independent random variables. Convolution product for normal, gamma, Poisson distributions. Maximum and minimum of independent random variables.
Limit Theorems. Markov's and Chebyshev's inequalities. Weak law of large numbers. Moment generating function and a brief proof of the Central Limit Theorem.
- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
- W. Feller, An introduction to probability theory and its applications (Wiley, 1968).
Programme
Combinatorial Analysis. Introduction to combinatorial calculations: permutations,combinations, examples.
Probability Axioms. Sample spaces, events, probability axioms. Equally likely events and other examples.
Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
Discrete Random Variables. Bernoulli, binomial, and Poisson random variables.
Poisson process. Other discrete distributions: geometric, hypergeometric, negative binomial. Expected value and variance of a discrete random variable. Examples.
Continuous Random Variables. Probability density function and distribution function. Uniform distribution on an interval, exponential, gamma, normal, Weibull, Cauchy distributions. Link between gamma distributions and Poisson process. Expected value and variance for continuous random variables.
Joint Distributions and Independent Random Variables. Joint distributions, independent random variables. Density of the sum of two independent random variables. Convolution product for normal, gamma, Poisson distributions. Maximum and minimum of independent random variables.
Limit Theorems. Markov's and Chebyshev's inequalities. Weak law of large numbers. Moment generating function and a brief proof of the Central Limit Theorem.
Core Documentation
- S. Ross, Calcolo delle probabilita' (Apogeo Ed.)- F. Caravenna e P. Dai Pra, Probabilita' (Springer Ed.)
- W. Feller, An introduction to probability theory and its applications (Wiley, 1968).
Type of delivery of the course
blackboard lecturesType of evaluation
final written exam and partial examinations