Discrete and continuous models: random variables, their distributions, examples and applications. Cumulative distribution functions, density functions, joint distributions. Conditional probability and independence. Density functions, moments, probability generating functions and characteristic functions. Approximation: strong law of large numbers, Central limit theorem. Statistics: sample, point estimation, confidence intervals, hypothesis testing. Markov chains, invariant probabilities.
-Baldi, Calcolo delle probabilita`
-S. Ross, Calcolo delle probabilita`
-Caravenna e Dai Pra, Probabilità
The course aims to provide the students with fundamental knowledge and understanding about discrete and continuous models and
random variables,. This is done in order to calculate probabilities needed in concrete situations and to calculate probability laws of one dimensional and multidimensional random variables in the case of known models and in the case in which there situation cannot be described via a known model. The course aims to provide the students with fundamental knowledge and understanding of limit theorems such as the Central limit theorem and the strong law of large numbers. One of the goals is to let the students develop basic technical skills, and critical thinking, needed when modeling concrete situations and solving non deterministic mathematical problems for which it is needed to estimate the required probabilities that cannot be exactly computed.
The course aims to explain how to use the knowledge of basic probability to understand basic notions in statistics. It wants to give knowledge and understanding of samples, estimates, point estimates, confidence intervals and hypothesis testing. Moreover, it aims to give different examples of applications and concrete situations where different tests have to be used in order to reject or not the null hypothesis, and where it is requested the computation of point or interval estimates.
The course aims to provide the students with fundamental knowledge and understanding about two basic examples of stochastic processes: Poisson process and Markov chains that are useful for Monte Carlo method. The course aims to let the students develop basic technical skills, and critical thinking, needed when modeling concrete situations and solving non deterministic problems involving stochastic processes.
Special attention will be paid to help the students to develop communication skills necessary to rigorously present the acquired knowledge in mathematical language and to justify clearly and precisely the solved exercises.
Derivation and interegration in one or more variables. Basic notions of geometry, linear algebra in order to be able to operate with matrices.
Lectures, discussion and correction of homework and statistical lab.
Type of Assessment
The exam consists of a written and oral examinations (under request) per session. In the written exam will have open questions in which the student has to show his knowledge and his ability to apply the results to solve problems and give their justification using formulas and the appropriate scientific language. Moreover, the questions will be formulated to highlight whether the student is able to choose the best probabilistic models to solve the concrete situation described in the exercise.
The oral examination is designed to evaluate the degree of understanding of the results presented in the course and the connections between different notions. In particular with the exposition of the definitions and results the teacher will verify the degree of comprehension of the notions and of the applied aspects. In the assessment, special attention is paid to communication skills, critical thinking and appropriate use of mathematical language.
Additionally, the students will have the opportunity to perform two partial tests that, if both successful, will allow them to access directly to the oral examination.
-Discrete models: discrete random variables, their distributions and distribution functions. Joint probability distributions and independence. expected values, probability models, variance and covariance. Particular distributions: Uniform, Bernoulli, binomial, negative binomial, geometric, hypergeometric, Poisson and multinomial. Law of large numbers, probability generating function.
-Continuous models: continuous random variables, their distributions, density function and distribution functions and properties. Joint probability distributions and independence. expected values, probability models, variance and covariance.
Law of large numbers. Normal, uniform and Gamma distributions. Waiting times and Poisson Process. Random generators, and simulations. Conditional expectation, probability generating function and Laplace transform or Characteristic function. Multivariate normal probability laws.
Convergence and approximation. Law of large numbers and Central Limit Theorem.
-Statistics: sample, estimators, point estimate, interval estimation.
- Markov chains, definitions and general properties, joint distributions, states classification, invariant probabilities, Metropolis algorithm, simulated annealing.