Probability Theory

How can randomness and uncertainty be described with mathematical precision? And why do many random processes lead to remarkably stable patterns and laws?

This textbook provides a systematic, mathematically rigorous, and accessible introduction to the fundamental concepts of probability theory. It is aimed primarily at mathematics students from the third semester onward, as well as students in computer science and other related fields.

The book covers random experiments, sample spaces and events, probability measures, random variables, important probability models, conditional probability, Bayes' theorem, independence, expectation, variance, and covariance. The Law of Large Numbers and the Central Limit Theorem are also developed and explained as fundamental results describing the behavior of random phenomena.

Alongside classical examples, the book presents a range of applications from computer science and machine learning. Numerous exercises with detailed, carefully structured solutions help readers apply the mathematical concepts confidently and develop a deeper understanding of the underlying theory.

A rigorous introduction to probability theory for university students, combining mathematical precision, clear explanations, and applications in computer science and machine learning.