Introduction to Probability, 2nd Edition

An intuitive, yet precise introduction to probability theory, stochastic processes, and probabilistic models used in science, engineering, economics, and related fields. The 2nd edition is a substantial revision of the 1st edition, involving a reorganization of old material and the addition of new material. The length of the book has increased by about 25 percent. The main new feature of the 2nd edition is thorough introduction to Bayesian and classical statistics. The book is the currently used textbook for ''Probabilistic Systems Analysis,'' an introductory probability course at the Massachusetts Institute of Technology, attended by a large number of undergraduate and graduate students. The book covers the fundamentals of probability theory (probabilistic models, discrete and continuous random variables, multiple random variables, and limit theorems), which are typically part of a first course on the subject, as well as the fundamental concepts and methods of statistical inference, both Bayesian and classical. It also contains, a number of more advanced topics, from which an instructor can choose to match the goals of a particular course. These topics include transforms, sums of random variables, a fairly detailed introduction to Bernoulli, Poisson, and Markov processes. The book strikes a balance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just intuitively explained in the text, but is developed in detail (at the level of advanced calculus) in the numerous solved theoretical problems. Written by two professors of the Department of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology, and members of the prestigious US National Academy of Engineering, the book has been widely adopted for classroom use in introductory probability courses within the USA and abroad.From a Review of the 1st Edition:...it trains the intuition to acquire probabilistic feeling. This book explains every single concept it enunciates. This is its main strength, deep explanation, and not just examples that happen to explain. Bertsekas and Tsitsiklis leave nothing to chance. The probability to misinterpret a concept or not understand it is just... zero. Numerous examples, figures, and end-of-chapter problems strengthen the understanding. Also of invaluable help is the book's web site, where solutions to the problems can be found-as well as much more information pertaining to probability, and also more problem sets. --Vladimir Botchev, Analog Dialogue Several other reviews can be found in the listing of the first edition of this book. Contents, preface, and more info at publisher's website (Athena Scientific, athenasc com) Title Page......Page 1 Preface......Page 4 Preface to the Second Edition......Page 7 Contents......Page 8 1. Sample Space and Probability......Page 11 1.1. Sets......Page 13 1.2. Probabilistic Models......Page 16 1.3. Conditional Probability......Page 28 1.4. Total Probability Theorem and Bayes' Rule......Page 38 1.5. Independence......Page 44 1.6. Counting......Page 54 1.7. Summary and Discussion......Page 61 Problems......Page 63 2. Discrete Random Variables......Page 81 2.1. Basic Concepts......Page 92 2.2. Probability Mass Functions......Page 94 2.3. Functions of Random Variables......Page 90 2.4. Expectation, Mean, and Variance......Page 91 2.5. Joint PMFs of Multiple Random Variables......Page 102 2.6. Conditioning......Page 107 2.7. Independence......Page 119 2.8. Summary and Discussion......Page 125 Problems......Page 129 3. General Random Variables......Page 149 3.1. Continuous Random Variables and PDFs......Page 150 3.2. Cumulative Distribution Functions......Page 158 3.3. Normal Random Variables......Page 163 3.4. Joint PDFs of Multiple Random Variables......Page 168 3.5. Conditioning......Page 174 3.6. The Continuous Bayes' Rule......Page 188 3.7. Summary and Discussion......Page 192 Problems......Page 194 4. Further Topics on Random Variables ......Page 210 4.1. Derived Distributions......Page 211 4.2. Covariance and Correlation......Page 226 4.3. Conditional Expectation and Variance Revisited......Page 231 4.4. Transforms......Page 238 4.5. Sum of a Random Number of Independent Random Variables ......Page 249 4.6. Summary and Discussion......Page 253 Problems......Page 255 5. Limit Theorems......Page 271 5.1. Markov and Chebyshev Inequalities......Page 273 5.2. The Weak Law of Large Numbers......Page 277 5.3. Convergence in Probability......Page 279 5.4. The Central Limit Theorem......Page 281 5.5. The Strong Law of Large Numbers......Page 288 5.6. Summary and Discussion......Page 290 Problems......Page 292 6. The Bernoulli and Poisson Processes......Page 303 6.1. The Bernoulli Process......Page 305 6.2. The Poisson Process......Page 317 6.3. Summary and Discussion......Page 332 Problems......Page 334 7. Markov Chains......Page 347 7.1. Discrete-Time Markov Chains......Page 348 7.2. Classification of States......Page 354 7.3. Steady-State Behavior......Page 360 7.4. Absorption Probabilities and Expected Time to Absorption......Page 370 7.5. Continuous-Time Markov Chains......Page 377 7.6. Summary and Discussion......Page 386 Problems......Page 388 8. Bayesian Statistical Inference......Page 414 8.1. Bayesian Inference and the Posterior Distribution......Page 419 8.2. Point Estimation, Hypothesis Testing, and the MAP Rule......Page 427 8.3. Bayesian Least Mean Squares Estimation......Page 437 8.4. Bayesian Linear Least Mean Squares Estimation......Page 444 8.5. Summary and Discussion......Page 451 Problems......Page 453 9. Classical Statistical Inference......Page 463 9.1. Classical Parameter Estimation......Page 468 9.2. Linear Regression......Page 483 9.3. Binary Hypothesis Testing......Page 492 9.4. Significance Testing......Page 502 9.5. Summary and Discussion......Page 511 Problems......Page 513 Index......Page 527 An intuitive, yet precise introduction to probability theory, stochastic processes, statistical inference, and probabilistic models used in science, engineering, economics, and related fields. This is the currently used textbook for "Probabilistic Systems Analysis," an introductory probability course at the Massachusetts Institute of Technology, attended by a large number of undergraduate and graduate students.The book covers the fundamentals of probability theory (probabilistic models, discrete and continuous random variables, multiple random variables, and limit theorems), which are typically part of a first course on the subject. It also contains, a number of more advanced topics, from which an instructor can choose to match the goals of a particular course. These topics include transforms, sums of random variables, a fairly detailed introduction to Bernoulli, Poisson, and Markov processes, Bayesian inference, and an introduction to classical statistics.The book strikes a balance between simplicity in exposition and sophistication in analytical reasoning. Some of the more mathematically rigorous analysis has been just intuitively explained in the text, but is developed in detail (at the level of advanced calculus) in the numerous solved theoretical problems "An intuitive, yet precise introduction to probability theory and probabilistic models used in science, engineering, economics, and related fields. Book features : develops the basic concepts of probability, random variables, stochastic processes, laws of large numbers, and the central limit theorem ; new in the 2nd Edition : two chapters on statistical inference, which allow the use of the book in combined probability and statistics courses ; illustrates the theory with many examples ; provides many theoretical problems that extend the book's coverage and enhance its mathematical foundation (solutions are included in the text) ; provides many exercises that enhance the understanding of the basic material (solutions are posted on the publisher's website) ; is supplemented by man unsolved practices exercises and other supportive class material posted on the publisher's website ; has been developed through extensive classroom use and experience."--taken from back cover 1. Sample Space And Probability -- 2. Discrete Random Variables -- 3. General Random Variables -- 4. Further Topics On Random Variables -- 5. The Bernoulli And Poisson Processes -- 6. Markov Chains -- 7. Limit Theorems. Dimitri P. Bertsekas And John N. Tsitsiklis. Includes Index.