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Probability theory is a discipline that studies the quantitative regularity of random phenomena. The fact that random phenomena arise, especially in the era of big data and artificial intelligence, determines the importance of this discipline. This volume introduces various concepts that quantitatively describe random phenomena, including probability, random variables, distribution functions, density functions, mathematical expectations, variances, moments, and characteristic functions. It finishes off by presenting probability limit theory, including various convergences.
Throughout the volume, great importance is attached to the elaboration of probability thoughts. For this reason, some practical examples to illustrate the introduced concept are always used. In order to meet the needs of different levels of readers, there is a section on Supplements and Notes at the end of each chapter to enhance and expand the content in the body of the textbook. This volume contains a large number of problems of varying levels for the reader with the purpose to review, consolidate, deepen and expand their knowledge.
As the only branch of mathematics that studies the quantitative regularity of random phenomena, probability theory has not only theoretical significance, but it is also a main theoretical basis of mathematical statistics. Therefore, it will be of interest to scholars from other disciplines related to random phenomena.
Contents:
- Events and Probabilities:
- Random Phenomena and Statistical Regularity
- Classical Probability Models
- The Axiomatic Definition of Probability
- Conditional Probability and Independent Events
- Random Variables and Distribution Functions:
- Discrete Random Variables
- Distribution Functions and Continuous Random Variables
- Random Vectors
- Conditional Distributions and Independence
- Functions of Random Variables
- Numerical Characteristics and Characteristic Functions:
- Mathematical Expectations
- Variances, Covariances and Correlation Coefficients
- Characteristic Functions
- Multivariate Normal Distributions
- Probability Limit Theorems:
- Convergence in Distribution and Central Limit Theorems
- Convergence in Probability and Weak Laws of Large Numbers
- Almost Sure Convergence and Strong Laws of Large Numbers
- Appendices:
- Distributions of Typical Random Variable
- Tables
Readership: Undergraduate and graduate students, and researchers interested in probability theory.
