# MATHEMATICAL AND STATISTICAL METHODS

Academic Year 2019/2020 - 3° Year - Curriculum A and Curriculum B
Teaching Staff: Orazio MUSCATO
Credit Value: 6
Scientific field: MAT/07 - Mathematical physics
Taught classes: 24 hours
Exercise: 24 hours
Term / Semester:

## Learning Objectives

The aim of the course is to provide a brief introduction to the statistical methodologies, the probability, the Monte Carlo method and the Markov chains. For this purpose, classical differential and integral calculation tools will be used, and, for applications, the electronic spreadsheet and MATLAB. The course is aimed at students enrolled in science degree courses (Computer Science, Mathematics, Physics, Engineering, etc.).

Learning Objectives

Knowledge and understanding: the aim of the course is to acquire knowledge that allows the student to analyze numerical data in order to make probabilistic forecasts.

Applying knowledge and understanding: the student will acquire the necessary skills to use commonly used statistical tools. In this regard, a part of the course will consist of lectures in the laboratory, with practical examples.

Making judjements: Through concrete examples, the student will be able to apply various statistical tests to better analyze the data.

Communication skills: the student will acquire the necessary communication skills and expressive appropriateness in the use of statistical language.

Learning skills: The course aims, as a goal, to provide the student with the necessary theoretical and practical methods to be able to face and solve new problems that may arise during a working activity. To this end, various topics will be treated in class involving the student in the search for possible solutions to real problems.

## Course Structure

Classroom lessons. Classroom exercise on spreadsheet.

## Detailed Course Content

Descriptive statistics. Numerical representations of statistical data. Graphic representations of frequency distributions. Trends, variability and shape. Linear and nonlinear regression for a set of data. Exercises with electronic spreadsheet.

Probability elements. Some probability definitions. Axiomatic definition of probability. Conditional probability. Bayes theorem. Discrete and continuous random variables. Central trend indices and variability.

Main distributions. Distribution of Bernoulli, Binomial, Poisson, Exponential, Weibull, Normal, Chi-square, Student. *Convergence theorems. Distribution Convergence, Large Number law, Central Limit Theorem.

Parameter estimates. Sampling and samples. Major sampling distributions. Timely estimators and estimates. Interval estimates: average confidence intervals and variance. Examples

Hypothesis Test. General characteristics of a hypothesis test. Parametric tests. Examples. Non-parametric tests. Test for the goodness of the adaptation. Kolmogorov-Smirnov's Test. Chi-square Test. Exercises with electronic spreadsheet.

Random numbers. Generators based on linear occurrences. Statistical tests for random numbers. Generating random numbers with assigned probability density: direct technique, rejection, combined.

The Monte Carlo method. Recall for Numerical Integration Methods. Monte Carlo Algorithm "Hit or Miss". Monte Carlo sampling algorithm. Monte-Carlo sample-mean algorithm. Variance reduction techniques: importance sampling, control variations, stratified sampling, antithetic variations. Direct Monte Carlo simulation for semiconductors.

Markov chains. Definitions and generalities. Calculation of joint laws. Classification of states. Invariant probabilities. Steady state. The Metropolis Algorithm. Basic about theory of queues

## Textbook Information

1. Professor's notes

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

3. P. Baldi, Calcolo delle probabilità e statistica , Mc Graw-Hill, Milano, 1992

4. M. Boella, Probabilità e statistica per ingegneria e scienze, Pearson Italia, 2010

5. F. Pelleray, Elementi di Statistica per le applicazioni, CELID, Torino