# Mathematical methods and statistics

Academic Year 2023/2024 - Teacher: GIOVANNI NASTASI

## Expected Learning Outcomes

The course aims to provide the main tools of probability and statistics together with the study of the Monte Carlo method and Markov chains. These tools make it possible to address questions concerning applications that are useful in the business and industrial fields and relevant to the cultural background of a computer science graduate. The objective of the course will be to develop problem-solving skills within the topics mentioned above. For this purpose, in support of mathematical techniques, the Python language will be used for computation.

## Course Structure

Frontal teaching. Classroom exercise with the help of the Python language.
If the course is taught in mixed or distance mode, the necessary changes may be introduced with respect to what was previously stated.

PLEASE NOTE: Information for students with disabilities and/or DSA.
To guarantee equal opportunities and in compliance with the laws in force, interested students can request a personal interview in order to plan any compensatory and/or dispensatory measures, based on the educational objectives and specific needs.

## Required Prerequisites

Elements of linear algebra and geometry. Elements of calculus. Elements of logic and combinatorics.

## Attendance of Lessons

Strongly recommended.

## Detailed Course Content

1. Elements of probability theory. Generalities and definitions of probability. Review of combinatorics. Probability of an event and properties. Conditional probability. Bayes theorem. Generalities of random variables. Distribution of a random variable and properties. Examples and exercises.
2. Discrete random variables. Generalities. Mean and variance of a discrete random variable. Bernoulli, binomial, hypergeometric, geometric, and Poisson distribution. Examples and exercises.
3. Continuous random variables. Generalities. Mean and variance of a continuous random variable. Uniform, normal, exponential, chi-square, Weibull's, and Student's distribution. Examples and exercises.
4. Law of large numbers and normal approximation. Convergence in probability. Law of large numbers. Convergence in law. Central limit theorem. Examples and exercises.
5. Descriptive and inferential statistics. Generalities. Grouping by individual values and by value classes. Punctual estimators. Confidence intervals for the mean and for the variance. Examples and exercises.
6. Hypothesis testing. General characteristics of a hypothesis test. Average test. Variance test. Nonparametric tests. Chi-square test. Kolmogorov-Smirnov test. Examples and exercises.
7. Linear regression. Generalities. Simple and multiple linear regression. Properties of the residuals and goodness of the linear regression model. Examples and exercises.
8. Pseudo-random numbers. Generalities. Random number generation with assigned probability density. Monte Carlo method for numerical integration. Examples and exercises.
9. Markov chains. Definitions and generalities. Calculation of joint laws. Classification of states. Invariant probabilities. Steady state. Examples and exercises.

## Textbook Information

1. V. Romano, Metodi matematici per i corsi di ingegneria, Città Studi, 2018.

2. P. Baldi, Stochastic Calculus. An Introduction Through Theory and Exercises, Springer Cham, 2017.

3. M.J. Evans, J.S. Rosenthal, Probability and Statistics: The Science of  Uncertainty, University of Toronto, lecture notes.
https://www.utstat.utoronto.ca/mikevans/research.html

4. D. C. Montgomery, G. C. Runger, Applied statistics and probability for engineers, 7th Edition, J. Wiley, 2018.

## Course Planning

SubjectsText References
1Elements of probability theory1, 2, 3
2Discrete random variables1, 2, 3
3Continuous random variables1, 2, 3
4Law of large numbers and normal approximation1, 2, 3
5Descriptive and inferential statistics.1, 3, 4
6Hypothesis testing1, 3, 4
7Linear regression1, 3, 4
8Pseudo-random numbers1, 2
9Markov chains1, 2

## Learning Assessment

### Learning Assessment Procedures

The exam consists of a written test followed, optionally, by a theoretical written test or by an oral test. The choice between the two options is left to the student. The written test will take place in the laboratory to allow the use of the Python language while solving the exercises. Those who score at least 18/30 pass the written test. For the calculation of the total mark, the written test based on exercises alone confers a score of a maximum of 24/30, the theoretical written test or the oral test a score of a maximum of 6/30. To take the theoretical test (written or oral) you must have passed the written test. The written theory test will be held on the same dates as the exercise-based written test.

The following parameters will be taken into account for the attribution of the final grade:
Score 29-30 cum laude: the student has an in-depth knowledge of probability theory tools, statistical investigation methods and stochastic models covered in the course, and is able to promptly and correctly integrate and critically analyze the situations presented, independently solve problems even of high complexity; He has excellent communication skills and language skills.
Score 26-28: the student has a good knowledge of probability theory tools, statistical investigation methods and stochastic models covered in the course, is able to integrate and analyze the situations presented in a critical and linear way, is able to solve in a quite autonomous in complex problems and explains the topics clearly using appropriate language.
Score 22-25: the student has a fair knowledge of probability theory tools, statistical investigation methods and stochastic models covered in the course, even if limited to the main topics; she manages to integrate and analyze the situations presented in a critical but not always linear way and expose the arguments quite clearly with a fair command of language.
Score 18-21: the student has minimal knowledge of probability theory tools, statistical investigation methods and stochastic models covered in the course, has a modest ability to integrate and critically analyze the situations presented and presents the topics clearly enough although the language skills are poorly developed.
Exam not passed: the student does not have the minimum required knowledge of the main contents of the course. The ability to use specific language is very poor or non-existent and he is not able to apply the acquired knowledge independently.

### Examples of frequently asked questions and / or exercises

Exercises on: probability calculation, parametric tests, chi-squared test, confidence intervals, linear regression, pseudo-random number generation.
Questions about: definition of probability, conditional probability, descriptive statistics, hypothesis testing, least squares method and linear regression, notable distributions and their properties, pseudo-random numbers, Monte Carlo method, and Markov chains.