# MATHEMATICAL STATISTICS

# MATHEMATICAL STATISTICS

Calculus, probability theory

The course aims at providing the students with advanced notions of mathematical statistics, by starting from the basic concepts of probability theory. After a brief review of probability theory, the first part of the course will focus on the solution of the main problems at the basis of estimation theory, both point-based and interval-based, and hypothesis testing. The main goal of this part of the course is to allow students to use estimation thery and hypothesis testing theory in practical application scenarios. The second part of the course will focus on the study of stochastic processes. Even in this case, the goal is teaching to the students how to apply the basic concepts of stochastic processes theory to typical problems of information engineering. To this end, in the last part of the course, the theoretical methods addressed in the first two parts will be applied to a number pf real-life problems including: detection and filtering of signals immersed in noise (a typical problem in radar and telecommuncation systems), biometric authentication based on iris analysis, and forensic analysis of digital images.

Part I: Basic concepts of Probability Theory

Axiomatic definition of probability

- Event space

Total probability and Bayes theorems

Joint events

- Conditional probabilities and independent events

Repeated trials

- Fundamental theorem of repeated trials and De Moivre – Laplace approximation

- Law of large numbers

Random variables

- Cumulative density function and distribution density function

- Some noticeable density functions

- Conditional density functions

- Expected values

- Joint density functions and independent random variables

Part II: Mathematical statistics

Estimation theory

- Unbiased and consistent estimators

- Point estimates

- Interval estimates

- Expected value and variance estimation

Robust estimation theory

Hypothesis testing

- First and second type errors

- ROC curve

- Bayesian test

- Neyman-Pearson criterion

Part III: Sthocastic processes

Some notes on signal and systems theory

- Finite energy and finite power signals

- Autocorrelation function

- Frequency analysis of finite power signals

- Time invariant linear systems: time and frequency analysis

Stochastic processes: main definitions

Temporal and ensemble analysis

Mean value and autocorrelation

Stationary stochastic processes

Frequency analysis of stochatic processes

- Power spectral density

- White noise

Stochastic processes and LTI systems

Ergodic processes

- Limits of sequences of random variables

- Mean and power ergodicity

- Spectral estimates

Gaussian processes

- Gaussian random variables

- Central limit theorem

- Gaussian processes and LTI systems

Part IV: Applications

Signal detection theory

- Signal detection in radar systems

- Matched filter

- Bayes and Neyman-Pearson criteria

Wiener filter

- Orthogonality principle

- Optimal Wiener filter

Biometric-based authentication

- Iris-based biometric authentication

- Iriscode

- Iris-based authentication as an hypothesis testing problem

Hypothesis testing for image forensics

- Image source identification

- Photo Response Non-Uniformity noise (PRNU)

- Source identification via PRNU

A. Papoulis, "Probability, Random Variables and Stochastic Processes", McGraw-Hill, 4th Edition

A. M. Kay, "Fundamentals of Statistical Signal Processing, Volume I: Estimation Theory", Prentic-Hall

A. M. Kay, "Fundamentals of Statistical Signal Processing, Volume II: Detection Theory", Prentic-Hall

Lecture notes

Lectures and exercises

Take-home project plus oral exam

Some lectures will be spent to recall the basic concepts of probability theory and signal analysis