Elementary probability theory: discrete distributions, repeated trials, continuous random variables. Some basic limit theorems and introduction to Markov chains.
Curriculum
scheda docente
materiale didattico
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.
Programma
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.
Testi Adottati
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Bibliografia Di Riferimento
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Modalità Frequenza
6 hours weeklyModalità Valutazione
Written examination and brief interview
scheda docente
materiale didattico
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.
Programma
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.
Testi Adottati
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Bibliografia Di Riferimento
William Feller, An introduction to probability theory and its applications. 3rd edition. Wiley, N.Y., (1968).Modalità Frequenza
6 hours weeklyModalità Valutazione
Written examination and brief interview