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Probability Theory

Book Description

Probability Theory is a classic topic in any course of exact sciences, that evolved from the amalgamation of different areas of mathematics, including set and measure theory. An axiomatic treatment of probability is presented in the book. Probability Theory is fundamental to several areas of knowledge, including engineering, computer science, mathematics, physics, sciences, economics, biology, medicine, social sciences and social communication. The book targets graduate students who may not have taken basic courses in these specific topics, and can provide a quick and concise way to obtain the knowledge they need to succeed in advanced courses.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. List of Figures
  5. Preface
  6. Acknowledgments
  7. 1. Advanced Set Theory
    1. 1.1 Set Theory
    2. 1.2 Basic Set Theory
    3. 1.3 The Axioms of Set Theory
    4. 1.4 Operations on Sets
    5. 1.5 Families of Sets
    6. 1.6 An Algebra of Sets
    7. 1.7 The Borel Algebra
  8. 2. Fundamentals Of Measure Theory
    1. 2.1 A Short History of Measure
    2. 2.2 Measure in an Algebra of Sets
    3. 2.3 The Riemann Integral
    4. 2.4 The Lebesgue Integral
  9. 3. Axiomatic Theory Of Probability
    1. 3.1 Basic Probability Theory
    2. 3.2 The Axioms of Probability
    3. 3.3 Bayes’Theorem
  10. 4. Random Variables
    1. 4.1 The Concept of a Random Variable
    2. 4.2 Cumulative Distribution Function
    3. 4.3 Moments of a Random Variable
    4. 4.4 Functions of Random Variables
    5. 4.5 Discrete Distributions
    6. 4.6 Characteristic Function
    7. 4.7 Conditional Distribution
    8. 4.8 Useful Distributions and Applications
  11. 5. Joint Random Variables
    1. 5.1 An Extension of the Concept of Random Variables
    2. 5.2 Properties of Probability Distributions
    3. 5.3 Moments in Two Dimensions
    4. 5.4 Conditional Moments
    5. 5.5 Two-Dimensional Characteristic Function
    6. 5.6 Function of Joint Random Variables
    7. 5.7 Complex Random Variables
  12. 6. Fundamental Inequalities
    1. 6.1 The World of Inequalities
    2. 6.2 Tchebychev’s Inequality
    3. 6.3 Markov’s Inequality
    4. 6.4 Bienaymé’s Inequality
    5. 6.5 Jensen’s Inequality
    6. 6.6 Chernoff’s Inequality
    7. 6.7 Kolmogorov’s Inequality
    8. 6.8 Schwarz’ Inequality
    9. 6.9 Hölder’s Inequality
    10. 6.10 Lyapunov’s Inequality
    11. 6.11 Minkowsky’s Inequality
    12. 6.12 About Arguments and Proofs
  13. 7. Convergence and the Law of Large Numbers
    1. 7.1 Forms of Convergence in Probability Theory
    2. 7.2 Types of Convergence
    3. 7.3 Relationships Between the Types of Convergence
    4. 7.4 Weak Law of Large Numbers
    5. 7.5 Strong Law of Large Numbers
    6. 7.6 Central Limit Theorem
  14. References
  15. Index
  16. Adpage