# MATHEMATICAL AND STATISTICAL METHODS

**Academic Year 2016/2017**- 3° Year

**Teaching Staff:**

**Orazio MUSCATO**

**Credit Value:**6

**Scientific field:**MAT/07 - Mathematical physics

**Taught classes:**48 hours

**Term / Semester:**1°

## Learning Objectives

To provide basic knowledge of statistics, probability, Monte carlo method and Markov chains.

## Detailed Course Content

**Descriptive statistics**. numerical statistical data representations. graphic representations of frequency distributions. indices of central tendency, variability and shape. linear and non-linear regression for a series of data. Exercises with spreadsheet.

**Elements of probability**. Some definitions of probability. Axiomatic definition of probability. Conditional probability. Bayes' theorem. Discrete and continuous random variables. Central tendency and variability indices.

**Important distributions**. Bernoulli, Binomial, Poisson, exponential, Weibull, Normal, Chi-square, Student. Convergence theorems. Convergence in distribution, law of large numbers, central limit theorem.

**Parameter Estimates**. Sampling and samples. Main sampling distributions. Estimators and point estimates. interval estimates: confidence intervals for the mean and variance. Examples

**Hypothesis testing**. General characteristics of a hypothesis test. Parametric Test. Examples. Nonparametric tests. Test for goodness of fit. Kolmogorov-Smirnov test. Test of Chi-Square. Exercises with spreadsheet.

**Random number generation**. Generators based on linear recurrences. Statistical tests for random numbers. Generation random numbers with assigned probability density: direct technique, rejection, combined.

**Monte Carlo method**. Recalls on numerical integration methods. Algorithm Monte Carlo "Hit or Miss". Algorithm Monte Carlo sampling. Algorithm Monte Carlo sample-mean. Techniques of variance reduction: importance sampling, control variates, stratified sampling, antithetic variates. Direct Simulation Monte Carlo for semiconductors.

**Markov chains**. Definitions and generalities. Joint laws calculus. Classification of states. Probability invariant. Steady state. Metropolis algorithm. Note on queuing theory.

## Textbook Information

notes provided by the teacher