Synopses & Reviews
Many probability books are written by mathematicians and have the built in bias that the reader is assumed to be a mathematician coming to the material for its beauty. This textbook is geared towards beginning graduate students from a variety of disciplines whose primary focus is not necessarily mathematics for its own sake. Instead, A Probability Path is designed for those requiring a deep understanding of advanced probability for their research in statistics, applied probability, biology, operations research, mathematical finance, and engineering.
Synopsis
This text is geared towards students who have no prior exposure to measure theory or advanced probability, and who require this material for their work in statistics, applied probability, operations research, mathematical finance, and engineering.
A one semester path is laid out in an efficient and disciplined way to cover the core material. The first three chapters provide a functioning knowledge of measure theory, Chapter 4 discusses independence, with expectation and integration covered in Chapter 5. This is followed by topics on different modes of convergence, laws of large numbers with applications to statistics (quantile and distribution function estimation) and applied probability. The remaining chapters offer a careful treatment of convergence in distribution and the central limit theorem. Finally, the text treats conditional expectation and martingales, closing on the fundamental theorem of mathematical finance.
Table of Contents
1. Sets and Events ; 2. Probability Spaces; 3. Random Variables, Elements and Measurable Maps; 4. Independence; 5. Integration and Expectation; 6. Convergence Concepts; 7. Laws of Large Numbers and Sums of Independent Random Variables; 8. Convergence in Distribution; 9. Characteristic Functions and the Central Limit Theorem; 10. Martingales; Index; References