## Probability and Statistics

The objective of the course Probability and Statistics is the development of basic skills for the modeling and mathematical analysis of processes with some kind of randomness. The development of probabilistic thinking is a fundamental component of modern scientific literacy. Its significance has been recognized in a variety of fields. Indeed, apart from the classical applications in the natural sciences, the probabilistic thinking is fundamental for the decision making in health-related issues, for the risk assessment in economic and actuarial problems etc. Moreover, Probability Theory is the prerequisite and foundation of Statistics which is nowadays used widely in the natural, social and economic sciences, as well as in biology and medicine. Probability and Statistics introduces the students in the basic probabilistic models and computational tools, by combining the mathematical approach with conceptual and intuitive understanding. Moreover, it contains a brief introduction to the Classical Statistical Inference, presenting some basic notions and techniques of point estimation, confidence intervals and hypothesis testing. The course is addressed to sophomore (undergraduate 2nd year) students in the Department of Informatics and Telecommunications at the University of Athens. More generally, it is accessible to students in departments of Mathematics, Statistics, the Natural Sciences and Engineering who have a basic background in Calculus.

### Objectives

The objective of the course is the development of basic skills for the modeling and mathematical analysis of processes and phenomena that include some kind of randomness. Moreover, it aims for the understanding of basic techniques for the statistical analysis of data. At the end of the course, the student is expected: • to model processes and situations from the every-day life and from other scientific disciplines in the framework of Probability Theory • to understand the basic concepts of Probability Theory, such as the concept of a sample point, the sample space, an event, a probability, a conditional probability, a random variable etc. • to be able to perform basic calculations of probabilities, expected values, variances etc. in problems that involve randomness. • to use conditioning arguments for the calculations of probabilities, expected values, variances etc. in problems that involve randomness. • to understand intuitively the basic convergence results of Probability Theory (laws of large numbers and the central limit theorem) and to be able to aply them for the approximate computations of probabilities. • to be able to use simple statistical data analyses, using basic techniques from point estimation, confidence intervals and hypotheses testing.

### Prerequisites

Knowledge of basic concepts and techniques from Calculus of one and several variables is a strict prerequisite for the course. In particular, the students should be familiar with computational techniques, such as the computation of integrals and sums. Some knowledge of Discrete Mathematics is desirable, though not necessary.

### Syllabus

The course aims to the development of basic skills in modeling and mathematical analysis of processes with some kind of randomness. Moreover, it contains an introduction to classical statistical inference. Its structure is as follows: • Sample space and events. Axiomatization of Probability Theory. • Conditional probability. • Total probability theorem and Bayes’ Rule. • Independence. • Basic counting principles. Finite sample spaces and classical probability. • Discrete random variables: Probability mass function, expectation and variance. • Multiple discrete random variables: Joint probability mass function, conditioning, independence. • Continuous random variables: Probability density function, expectation and variance. • Multiple continuous random variables: Joint probability density function, conditioning, independence. • Distributions of functions of random variables. • Covariance and correlation. • Conditional expectation and variance. • Law of Large Numbers and Central Limit Theorem. • Introduction to the classical statistical inference. • Introduction to point estimation and confidence intervals. • Introduction to hypothesis testing.

COURSE DETAILS
 Level: Type: Undergraduate (A+) Instructors: Antonis Economou Department: Department of Informatics and Telecommunications Institution: National and Kapodistrian University of Athens Subject: Mathematics Rights: CC - Attribution-NonCommercial-ShareAlike