Probability Theory and Mathematical Statistics

In the textbook, the mathematical foundations of the probability theory are presented on the basis of Kolmogorov's axiomatics. In the first chapter, materials on random events and their probabilities are presented in the framework of a discrete probability space. The second chapter is devoted to the general probability space. In the third chapter, a random variable is defined as a measurable function. The concept of mathematical expectation considered in the fourth chapter is introduced as a Lebesgue integral in a probability measure on a general probability space, while the readers are not required to have any information about the Lebesgue integral. The fifth chapter is devoted to limit theorems in the Bernoulli scheme. The last sixth and seventh chapters contain materials on various types of convergence of sequences of random variables and on the laws of large numbers. The textbook is recommended for undergraduate, graduate and doctoral PhD students studying in mathematical specialties and specialties related to mathematics.