Probability and Mathematical Statistics 1

A.Y. 2016/2017
Overall hours
Learning objectives
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.
Course syllabus and organization

Single session

Lesson period
Second semester
Course syllabus
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 Conditional Probability 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 The Binomial Variable Waiting Times The geometric variable The negative binomial variable 1.6.2. The Poisson process The Poisson Variable Waiting Times The exponential variable 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 Data: frequencies, diagrams and histograms.
3.1.2 Data synthesis. Main statistical indexes: sample mean, mode, median, sample variance, range, sample standard deviation, kurtosis, asymmetry,. Percentiles, quartiles and box-plot
3.1.3 Bivariate Data
3.2. Sample Simulation
3.2.1 Inverse transform method. Simulation of an exponential distribution Simulation of a discrete random variable Simulation of a Binomial distribution Simulation of a Poisson distribution 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
MAT/06 - PROBABILITY AND STATISTICS - University credits: 9
Practicals: 33 hours
Laboratories: 24 hours
Lessons: 36 hours
Professors: Potrich Norman, Ugolini Stefania, Villa Elena
Professors: Potrich Norman, Villa Elena
Turno A
Professor: Potrich Norman
Turno B
Professor: Ugolini Stefania
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