Probability Theory

1. Front Matter

   Pages i-xv

   PDF

2. Classes of Sets, Measures, and Probability Spaces

   • Yuan Shih Chow, Henry Teicher

   Pages 1-29

3. Binomial Random Variables

   • Yuan Shih Chow, Henry Teicher

   Pages 30-53

4. Independence

   • Yuan Shih Chow, Henry Teicher

   Pages 54-82

5. Integration in a Probability Space

   • Yuan Shih Chow, Henry Teicher

   Pages 83-109

6. Sums of Independent Random Variables

   • Yuan Shih Chow, Henry Teicher

   Pages 110-155

7. Measure Extensions, Lebesgue-Stieltjes Measure, Kolmogorov Consistency Theorem

   • Yuan Shih Chow, Henry Teicher

   Pages 156-197

8. Conditional Expectation, Conditional Independence, Introduction to Martingales

   • Yuan Shih Chow, Henry Teicher

   Pages 198-246

9. Distribution Functions and Characteristic Functions

   • Yuan Shih Chow, Henry Teicher

   Pages 247-289

10. Central Limit Theorems

    • Yuan Shih Chow, Henry Teicher

    Pages 290-323

11. Limit Theorems for Independent Random Variables

    • Yuan Shih Chow, Henry Teicher

    Pages 324-373

12. Martingales

    • Yuan Shih Chow, Henry Teicher

    Pages 374-411

13. Infinitely Divisible Laws

    • Yuan Shih Chow, Henry Teicher

    Pages 412-445

14. Back Matter

    Pages 446-458

    PDF

Probability theory is a branch of mathematics dealing with chance phenomena and has clearly discernible links with the real world. The origins of the sub­ ject, generally attributed to investigations by the renowned french mathe­ matician Fermat of problems posed by a gambling contemporary to Pascal, have been pushed back a century earlier to the italian mathematicians Cardano and Tartaglia about 1570 (Ore, 1953). Results as significant as the Bernoulli weak law of large numbers appeared as early as 1713, although its counterpart, the Borel strong law oflarge numbers, did not emerge until 1909. Central limit theorems and conditional probabilities were already being investigated in the eighteenth century, but the first serious attempts to grapple with the logical foundations of probability seem to be Keynes (1921), von Mises (1928; 1931), and Kolmogorov (1933). An axiomatic mold and measure-theoretic framework for probability theory was furnished by Kolmogorov. In this so-called objective or measure­ theoretic approach, definitions and axioms are so chosen that the empirical realization of an event is the outcome of a not completely determined physical experiment -an experiment which is at least conceptually capable of indefi­ nite repetition (this notion is due to von Mises). The concrete or intuitive counterpart of the probability of an event is a long run or limiting frequency of the corresponding outcome.

• Probability
• Probability theory
• Wahrscheinlichkeit
• Wahrscheinlichkeitsrechnung
• law of large numbers

• Department of Mathematics and Statistics, Columbia University, New York, USA

  Yuan Shih Chow

• Department of Statistics, Rutgers University, New Brunswick, USA

  Henry Teicher

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