Ebook: Probability Essentials

This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference.

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This introduction to Probability Theory can be used, at the beginning graduate level, for a one-semester course on Probability Theory or for self-direction without benefit of a formal course; the measure theory needed is developed in the text. It will also be useful for students and teachers in related areas such as Finance Theory (Economics), Electrical Engineering, and Operations Research. The text covers the essentials in a directed and lean way with 28 short chapters. Assuming of readers only an undergraduate background in mathematics, it brings them from a starting knowledge of the subject to a knowledge of the basics of Martingale Theory. After learning Probability Theory from this text, the interested student will be ready to continue with the study of more advanced topics, such as Brownian Motion and Ito Calculus, or Statistical Inference.
Content:
Front Matter....Pages I-X
Introduction....Pages 1-3
Axioms of Probability....Pages 5-10
Conditional Probability and Independence....Pages 11-16
Probabilities on a Countable Space....Pages 17-19
Random Variables on a Countable Space....Pages 21-29
Construction of a Probability Measure....Pages 31-33
Construction of a Probability Measure on R....Pages 35-42
Random Variables....Pages 43-46
Integration with Respect to a Probability Measure....Pages 47-60
Independent Random Variables....Pages 61-72
Probability Distributions on R....Pages 73-81
Characteristic Functions....Pages 83-97
Properties of Characteristic Functions....Pages 99-105
Sums of Independent Random Variables....Pages 107-112
Gaussian Random Variables (The Normal and the Multivariate Normal Distributions)....Pages 113-119
Convergence of Random Variables....Pages 121-135
Weak Convergence....Pages 137-145
Weak Convergence and Characteristic Functions....Pages 147-162
The Laws of Large Numbers....Pages 163-167
The Central Limit Theorem....Pages 169-175
Conditional Expectation....Pages 177-184
Martingales....Pages 185-191
Supermartingales and Submartingales....Pages 193-205
Martingale Inequalities....Pages 207-213
Martingale Convergence Theorems....Pages 215-218
The Radon-Nikodym Theorem....Pages 219-223
Back Matter....Pages 225-238
....Pages 239-243