Elements of distribution theory

The 2006 INFORMS Expository Writing Award-winning and best-selling author Sheldon Ross (University of Southern California) teams up with Erol Peköz (Boston University) to bring you this textbook for undergraduate and graduate students in statistics, mathematics, engineering, finance, and actuarial science. This is a guided tour designed to give familiarity with advanced topics in probability without having to wade through the exhaustive coverage of the classic advanced probability theory books. Topics include measure theory, limit theorems, bounding probabilities and expectations, coupling and Stein's method, martingales, Markov chains, renewal theory, and Brownian motion. No other text covers all these advanced topics rigorously but at such an accessible level; all you need is calculus and material from a first undergraduate course in probability "This detailed introduction to distribution theory is designed as a text for the probability portion of the first year statistical theory sequence for Master's and PhD students in statistics, biostatistics and econometrics. The text uses no measure theory, requiring only a background in calculus and linear algebra. Topics range from the basic distribution and density functions, expectation, conditioning, characteristic functions, cumulants, convergence in distribution and the central limit theorem to more advanced concepts such as exchangeability, models with a group structure, asymptotic approximations to integrals and orthogonal polynomials. An appendix gives a detailed summary of the mathematical definitions and results that are used in the book"--Book cover. Properties of probability distributions -- 2. Conditional distributions and expectation -- 3. Characteristic functions -- 4. Moments and cumulants -- 5 Parametric families of distributions -- 6. Stochastic processes -- 7. Distribution theory for functions of random variables -- 8. Normal distribution theory -- 9. Approximation of integrals -- 10. Orthogonal polynomials -- 11 Approximation of probability distributions -- 12. Central limit theorems -- 13. Approximations to the distributions of more general statistics -- 14. Higher-order asymptotic approximations -- App. 1 Integration with respect to a distribution function -- App. 2 Basic properties of complex numbers