The course aims to provide an introduction to the methods for modeling random phenomena and for the statistical analysis of experimental data.
Expected learning outcomes
The student, on the basis of concrete examples arising also from everyday experience, is led to the construction of mathematical models for random phenomena based on an axiomatic system due to Kolmogorov. The analysis of the relevant mathematical structures, including random variables, their distributions and limit theorems, is faced. Moreover the course aims to introduce basic statistical methods, including parameter estimation, computation of confidence intervals, and hypothesis testing. The lab is intended to guide each student to learn by doing, through simulations of probabilistic models and the calculation of estimators for unknown parameters, based on the actual or simulated samples. Basic methods of descriptive statistics will be also discussed, in order to make easier the connection with everyday experience.
1. Probabilistic Models 1.1 Introduction to Probability Theory 1.1.1 Sample space and events 1.1.2 Algebras and sigma-algebras 1.1.3 Probability Measures 1.1.4 Construction of a Probability Space 1.1.5 Conditioning 18.104.22.168 Conditional Probability 22.214.171.124 Prior and Posterior Probability. Bayes Formula 1.1.6 Stochastic or Statistical Independence 1.2 Random Variables 1.2.1 The notion of random variable 1.2.2 Probability Measures induced by Random Variables 1.2.3 Discrete Random Variables 1.2.4 Random Vectors 1.2.5 Dichotomic Experiments. The Bernoulli Scheme 1.3. Random Variables and their distributions 1.4. Moments of Random Variables 1.5. Correlation and independence 1.6. Stochastic Processes 1.6.1 The Bernoulli process 126.96.36.199. The Binomial Variable 188.8.131.52. Waiting Times 184.108.40.206.1. The geometric variable 220.127.116.11.2. The negative binomial variable 1.6.2. The Poisson process 18.104.22.168. The Poisson Variable 22.214.171.124. Waiting Times 126.96.36.199.1. The exponential variable 188.8.131.52.2. The gamma variable 1.7. The (weak) law of large numbers. 1.8. The central limit theorem 1.8.1. Sums of independent random variables 1.8.2. The normal variable 1.8.3. The chi-square variable 1.8.4. The Student variable 1.8.5. Approximation of discrete variables 1.9. Random Vectors 1.9.1 Gaussian Vectors 1.10. Transformations of random variables 1.11. Conditional Distributions 1.11.1 Conditional Expected Value 2. Samples and Statistics 2.1. Descriptive Statistics. Samples and histogram 2.2. Distributions of the fundamental statistics 2.3. Estimation 2.3.1. Parameter estimation 2.3.2. Interval estimations 2.3.3. Maximum likelihood method 2.3.4. Sample dimension 3. Lab 3.1. Descriptive Statistics 3.1.1 Explorative Analysis of 1dim data 184.108.40.206 Data: frequencies, diagrams and histograms. 3.1.2 Data synthesis. 220.127.116.11 Main statistical indexes: sample mean, mode, median, sample variance, range, sample standard deviation, kurtosis, asymmetry,. 18.104.22.168 Percentiles, quartiles and box-plot 3.1.3 Bivariate Data 3.2. Sample Simulation 3.2.1 Inverse transform method. 22.214.171.124 Simulation of an exponential distribution 126.96.36.199 Simulation of a discrete random variable 188.8.131.52 Simulation of a Binomial distribution 184.108.40.206 Simulation of a Poisson distribution 220.127.116.11 Simulation of a Geometric Distribution 3.2.2 Rejection method 3.2.3 Simulation of two independent gaussian 3.3 Simulation of a stochastic process 3.3.1 Bernoulli process 3.3.2 Binomial process 3.3.3 Poisson process