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. Axioms of Probability. Sample spaces, events, probability axioms. Equiprobable events and further examples.
3. Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
4. Discrete Random Variables.** Bernoulli, binomial, and Poisson random variables. The Poisson process. Other discrete distributions: geometric, hypergeometric, and negative binomial. Expectation and variance of discrete random variables. Examples.
5. Continuous Random Variables. Probability density functions and cumulative distribution functions. Uniform, exponential, gamma, Gaussian (normal), Weibull, and Cauchy distributions. Relationship between the gamma distribution and the Poisson process. Expectation and variance of continuous random variables. The transformation method for simulating continuous random variables.
6. Joint Distributions and Independent Random Variables. Joint distributions and independent random variables. Distribution of the sum of two independent random variables. Convolution for normal, gamma, and Poisson distributions. Maxima and minima of independent random variables.
7. Limit Theorems. Markov's and Chebyshev's inequalities. Weak Law of Large Numbers. Moment generating functions and an outline of the proof of the Central Limit Theorem.
Programme
1. Combinatorial Analysis. Introduction to combinatorics: permutations, combinations, and examples.2. Axioms of Probability. Sample spaces, events, probability axioms. Equiprobable events and further examples.
3. Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
4. Discrete Random Variables.** Bernoulli, binomial, and Poisson random variables. The Poisson process. Other discrete distributions: geometric, hypergeometric, and negative binomial. Expectation and variance of discrete random variables. Examples.
5. Continuous Random Variables. Probability density functions and cumulative distribution functions. Uniform, exponential, gamma, Gaussian (normal), Weibull, and Cauchy distributions. Relationship between the gamma distribution and the Poisson process. Expectation and variance of continuous random variables. The transformation method for simulating continuous random variables.
6. Joint Distributions and Independent Random Variables. Joint distributions and independent random variables. Distribution of the sum of two independent random variables. Convolution for normal, gamma, and Poisson distributions. Maxima and minima of independent random variables.
7. Limit Theorems. Markov's and Chebyshev's inequalities. Weak Law of Large Numbers. Moment generating functions and an outline of the proof of the Central Limit Theorem.
Core Documentation
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Reference Bibliography
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Attendance
6 hours weeklyType of evaluation
Written examination and brief interview 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
2. Axioms of Probability. Sample spaces, events, probability axioms. Equiprobable events and further examples.
3. Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
4. Discrete Random Variables.** Bernoulli, binomial, and Poisson random variables. The Poisson process. Other discrete distributions: geometric, hypergeometric, and negative binomial. Expectation and variance of discrete random variables. Examples.
5. Continuous Random Variables. Probability density functions and cumulative distribution functions. Uniform, exponential, gamma, Gaussian (normal), Weibull, and Cauchy distributions. Relationship between the gamma distribution and the Poisson process. Expectation and variance of continuous random variables. The transformation method for simulating continuous random variables.
6. Joint Distributions and Independent Random Variables. Joint distributions and independent random variables. Distribution of the sum of two independent random variables. Convolution for normal, gamma, and Poisson distributions. Maxima and minima of independent random variables.
7. Limit Theorems. Markov's and Chebyshev's inequalities. Weak Law of Large Numbers. Moment generating functions and an outline of the proof of the Central Limit Theorem.
Programme
1. Combinatorial Analysis. Introduction to combinatorics: permutations, combinations, and examples.2. Axioms of Probability. Sample spaces, events, probability axioms. Equiprobable events and further examples.
3. Conditional Probability and Independence. Conditional probability, Bayes' theorem, independent events.
4. Discrete Random Variables.** Bernoulli, binomial, and Poisson random variables. The Poisson process. Other discrete distributions: geometric, hypergeometric, and negative binomial. Expectation and variance of discrete random variables. Examples.
5. Continuous Random Variables. Probability density functions and cumulative distribution functions. Uniform, exponential, gamma, Gaussian (normal), Weibull, and Cauchy distributions. Relationship between the gamma distribution and the Poisson process. Expectation and variance of continuous random variables. The transformation method for simulating continuous random variables.
6. Joint Distributions and Independent Random Variables. Joint distributions and independent random variables. Distribution of the sum of two independent random variables. Convolution for normal, gamma, and Poisson distributions. Maxima and minima of independent random variables.
7. Limit Theorems. Markov's and Chebyshev's inequalities. Weak Law of Large Numbers. Moment generating functions and an outline of the proof of the Central Limit Theorem.
Core Documentation
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Reference Bibliography
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Attendance
6 hours weeklyType of evaluation
Written examination and brief interview 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).