Mathematical methods and statistics A - L

General learning objectives in terms of expected learning outcomes.

1. Knowledge and understanding: The purpose of the course is to acquire knowledge that will enable the student to understand the basics of probability and statistics and of Monte Carlo methods and Markov chain. The course also provides some basics of Python language.
2. Ability to apply knowledge and understanding: the student will acquire the skills needed to face some applicative problems of great utility in industrial and factory fields. The acquired kowledge will be tested through python programming.
3. Making judgments: Through examples in the classroom, the student will be put into the condition of understanding whether the solutions offered by him meet a certain degree of quality. The double check with numerical experiments will provide a quick confirmation of the acquired knowledges.
   
4. Communication skills: The student will acquire the necessary communication skills and technical language skills in the probability and statistical field.
5. Learning Skills: The theoretical and practical methodologies provided by the course allow the student to deal with amore sofisticate problems in the field of probability and statistics

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.

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.

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

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

• 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
  

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

• Lecture notes

The exam consists of two mandatory parts, a laboratory part and a theory part, to be completed in a single session and evaluated separately. 

The laboratory part aims to evaluate the student's ability to apply in practice the concepts learned during the course. During this part the student will have to solve two exercises using the Python language, and the elaborate must be dressed with illustrative comments on the functioning of the code produced. 

The theory part aims to evaluate the notions learned by the student during the course. During this part the student will have to answer in writing to two theory questions. 

The test will last two hours and will be held in the laboratory. The student will be able to independently decide how to divide the time available between the two parts. During the exam it will be possible to bring only a formulary, previously approved by the teacher. 

Each test will be assigned a grade. Each lab exercise will have a maximum score of 16. The total score that can be achieved in the lab part is 32. Each theory question will have a maximum score of 16. The total score that can be achieved in the theory part is 32. The test is passed by those who achieve 18 in both parts. The final grade will be the arithmetic mean, expressed in thirtieths, between the theory grade and the lab grade. If this mean exceeds 30, 30 cum laude will be recorded.

The following parameters will be taken into account for the attribution of the final grade: