Fundamentals of Probability: A First Course

Probability theory is one branch of mathematics that is simultaneously deep and immediately applicable in diverse areas of human endeavor. It is as fundamental as calculus. Calculus explains the external world, and probability theory helps predict a lot of it. In addition, problems in probability theory have an innate appeal, and the answers are often structured and strikingly beautiful. A solid background in probability theory and probability models will become increasingly more useful in the twenty-?rst century, as dif?cult new problems emerge, that will require more sophisticated models and analysis. Thisisa text onthe fundamentalsof thetheoryofprobabilityat anundergraduate or ?rst-year graduate level for students in science, engineering,and economics. The only mathematical background required is knowledge of univariate and multiva- ate calculus and basic linear algebra. The book covers all of the standard topics in basic probability, such as combinatorial probability, discrete and continuous distributions, moment generating functions, fundamental probability inequalities, the central limit theorem, and joint and conditional distributions of discrete and continuous random variables. But it also has some unique features and a forwa- looking feel.

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This is a text encompassing all of the standard topics in introductory probability theory, together with a significant amount of optional material of emerging importance. The emphasis is on a lucid and accessible writing style, mixed with a large number of interesting examples of a diverse nature. The text will prepare students extremely well for courses in more advanced probability and in statistical theory and for the actuary exam.

The book covers combinatorial probability, all the standard univariate discrete and continuous distributions, joint and conditional distributions in the bivariate and the multivariate case, the bivariate normal distribution, moment generating functions, various probability inequalities, the central limit theorem and the laws of large numbers, and the distribution theory of order statistics. In addition, the book gives a complete and accessible treatment of finite Markov chains, and a treatment of modern urn models and statistical genetics. It includes 303 worked out examples and 810 exercises, including a large compendium of supplementary exercises for exam preparation and additional homework. Each chapter has a detailed chapter summary. The appendix includes the important formulas for the distributions in common use and important formulas from calculus, algebra, trigonometry, and geometry.

Anirban DasGupta is Professor of Statistics at Purdue University, USA. He has been the main editor of the Lecture Notes and Monographs series, as well as the Collections series of the Institute of Mathematical Statistics, and is currently the Co-editor of the Selected Works in Statistics and Probability series, published by Springer. He has been an associate editor of the Annals of Statistics, Journal of the American Statistical Association, Journal of Statistical Planning and Inference, International Statistical Review, Sankhya, and Metrika. He is the author of Asymptotic Theory of Statistics and Probability, 2008, and of 70 refereed articles on probability and statistics. He is a Fellow of the Institute of Mathematical Statistics.

Contenu
Introducing Probability.- The Birthday and Matching Problems.- Conditional Probability and Independence.- Integer-Valued and Discrete Random Variables.- Generating Functions.- Standard Discrete Distributions.- Continuous Random Variables.- Some Special Continuous Distributions.- Normal Distribution.- Normal Approximations and the Central Limit Theorem.- Multivariate Discrete Distributions.- Multidimensional Densities.- Convolutions and Transformations.- Markov Chains and Applications.- Urn Models in Physics and Genetics.