The goal of the course is to introduce the tools of probability, mathematical statistics and mathematical analysis for producing, selecting and processing information and for time-frequency signals analysis. The students learn the methodology and the logic for modeling and evaluating random phenomena, data analysis and statistical inference, and also the basic techniques for the computation of the transforms for impulsive phenomena analysis. Students are expected to acquire the skills to solve the most frequent practical problems. Students' mastery of concepts as well as the ability to apply them will be ascertained through discussions and in-class exercises.

Students are required to be familiar with the notions and tools of the mathematics courses of the first year and of the first semester of the second year: these include differential and integral calculus of one or several variables and complex variables functions theory.

1. Descriptive statistics: frequency tables and graphs. Central tendency and variability indexes, percentiles and correlation coefficient (5h). 2. Elementary probability: elements of combinatorics; probability definitions; conditional probability, independence (15h). 3. Discrete and absolutely continuous random variables: distributions, expected values and variances. Notable examples (15h). 4. Jointly distributed random variables: marginals and conditional distributions. Independence, correlation e conditional independence, notable multivariate distributions (15h). 5. Convergence in probability and in distribution: weak law of large numbers and central limit theorem (5h). 6. Statistical inference: sampling from a population, sampling statistics and their distribution; parameter estimation, estimators and related properties, maximum likelihood estimators, pseudo-random numbers and examples of related generating alghoritms (10h). 7. Confidence intervals: confidence intervals for the mean and for proportions, asymptotic confidence intervals. Basics of hypothesis testing ​(10h). 8. Multiple linear regression and least square estimation (5h). 9. Theory of distributions: definitions and basic operations (algebraic operations, translation, rescaling, derivatives), Dirac delta, p.v.(1/t), Dirac comb. Convolution of functions and distributions (8h). 10. Fourier and Laplace transforms of complex valued functions and of distributions: definitions and properties, inverse transforms, inversion formula. Notable transforms (12h).

Exercises will cover the topics of the lectures. Some of them will be carried out by the instructor at the blackboard, others will actively involve the students.

Lecture notes will be available in the course web page. Recommended textbook in probability and mathematical statistics: Ross, S. 'Probability and Statistics for Engineers and Scientists', Academic Press 5th Ed., 2014 (or any other edition). For analysis, lecture notes and texts will be suggested by the instructor.

Lecture slides; Exercises; Exercise with solutions ; Video lectures (previous years); Self-assessment tools;

You can take this exam before attending the course

Exam: Written test; Optional oral exam;

Assessment methods: Written individual open and/or closed answers questions closed books exam. The oral exam is optional on students' request or at the discretion of the instructor. The exam is 2 hour long. Students are allowed to use only a pen, white paper, the calculator and the formulae sheets provided by the instructors and available on the course website. The exam is composed of two parts: 1. ten multiple-choice quizzes, two of which in analysis, four in probability and four in statistics; 2. three exercises, one in analysis, one in probability and one in statistics (each composed of different questions). For each quiz, four possible answers are shown, only one of which is correct. Each answer to the test is evaluated 1 point if correct and 0 otherwise. Thus the maximum score to the test is 10. The goal of the multiple choice test is to verify the understanding of the fundamental basic concepts of each of the course modules. The goal of these exercises is to verify the knowledge and understanding of the students main tools for solving the problems taught during the lectures and practice sessions. Each of the three exercises is composed . The maximum score of the analysis exercise is 5 points, whereas the maximum score of the probability and statistics exercises is 9 points each. If the sum of the two parts scores of the exam is less or equal to 30, it represents the final mark. If it is 31, the final mark is 30 and, if it is 31 or 32, the final mark is 30 with honor (30L). An oral exam can be required at the discretion of the instructor in the case a further investigation is needed to ascertain students' mastery of the concepts delivered in the course. Only students who passed the exam can ask to be admitted to the oral exam. In particular, if an oral exam is asked and performed, it becomes part of the evaluation together with the written part. Depending on the performance of the student, the final mark could be less, equal or greater than the total score of the written exam.

In addition to the message sent by the online system, students with disabilities or Specific Learning Disorders (SLD) are invited to directly inform the professor in charge of the course about the special arrangements for the exam that have been agreed with the Special Needs Unit. The professor has to be informed at least one week before the beginning of the examination session in order to provide students with the most suitable arrangements for each specific type of exam.