# MATHEMATICAL AND STATISTICAL METHODS

Academic Year 2017/2018 - 3° Year
Teaching Staff: Orazio MUSCATO
Credit Value: 6
Scientific field: MAT/07 - Mathematical physics
Taught classes: 48 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.).

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

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

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

5. S.M. Ross, Probabilità e statistica, Apogeo, Milano