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MIT 6 041 - FUNDAMENTALS OF APPLIED PROBABILITY THEORY

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FUNDAMENTALS OF APPLIED PROBABILITY THLQRYALVIN W. DRAKE Operations Research Center and Department of Electrical Engineering Massachusetts Institute of Technology a A m McGRAW-HILL CLASSIC TEXTBci'K REISSUE McGraw-Hill Classic Textbook Reissue Series AMYX, BASS and WHITING: Petroleum Reservoir Engineering: Physical Properties fundamentalsCHOW: Open-Channel ~~draulics DAVENPORT: Probability Random Process: An Introduction for Applied Scientists and Engineers of DRAKE: Fundamentals of Applied Probability Theory GOODMAN: Introduction to Fourier Optics appliedHARRINGTON: Time-Harmonic Electromagnetic Fields HINZE: Turbulence probabilityKAYS and CRAWFORD: Convective Heat and Mass Transfer KRYNINE and JUDD: Principles of Engineering Geology and Geotechnics MEIROVITCH: Methods of Analytical Dynamics theory MELSA and SCHULTZ: Linear Control Systems MICKLEY, SHERWOOD and REED: Applied Mathematics in Chemical Engineering PAPOULIS: The Fourier Integral and Its Applications PHELAN: Fundamentals of Mechanical Design SCHLICHTING:. Boundary Layer Theory SCHWARTZ and SHAW: Signal Processing: Discrete Spectral Analysis, Detection, and Estimation TIMOSHENKO: Theory of Plates and Shells TIMOSHENKO and GOODIER: Theory of Elasticity TIMOSHENKO and GERE: Theory of Elastic Stability ' TREYBAL: Mass-Transfer Operations TRUXAL: Introductory Systems Engineering McGraw-Hill, Ine.WARNER and McNEARY: Applied Descriptive Geometry New York St. Louis San Francisco Auckland Bog06 WELLMAN: Technical Descriptive Geometry Caracas Lisbon London Madrid Mexico City Milan Montreal New Delhi San Juan Singapore Sydney Tokyo TorontoPREFACE This is a first textbook in applied probability theory, assuming a background of one year of calculus. The material represents a one- semester subject taught at M.I.T. to about 250 students per year, most of whom are in the Schools of Engineering or Management. About two-thirds of these students are undergraduates. The subject, Probabilistic Systems Analysis, serves both as a terminal course and as a prerequisite for more advanced work in areas such as communica- tion theory, control systems, decision theory, operations research, quantitative management, statistics, and stochastic processes.ix viii PREFACE My intention is to present a physically based introduction to applied probability theory, with emphasis on the continuity of funda- mentals. A prime objective isto develop in the new student an under- standing of the nature, formulation, and analysis of probabilistic situations. This text stresses the sample space of representation of probabilistic processes and (especially in the problems) the need for explicit modeling of nondeterministic processes. In the attempt to achieve these goals, several traditional details have either been omitted or relegated to an appendix. Appreciable effort has been made to avoid the segmentation and listing of detailed applications which must appear in a truly comprehensive work in this area. Intended primarily as a student text, this book is not suitable for use as a general reference. Scope'and Organization The fundamentals of probability theory, beginning with a dis-cussion of the algebra of events and concluding with Bayes' theorem, are presented in Chapter 1. An axiomatic development of probability theory is used and, wherever possible, concepts are interpreted in the sample space representation of the model of an experiment (any non- deterministic process). The assignment of probability measure in the modeling of physical situations is not necessarily tied to a relative frequency interpretation. In the last section of this chapter, the use of sample and event spaces in problems of enumeration is demonstrated. Chapter 2 is concerned with the extension of earlier results to deal with random variables. .This introductory text emphasizes the local assignment of probability in sample space. For this reason, we work primarily with probability density functions rather than cumula- tive distribution functions. My experience is that this approach is much more intuitive for the beginning student. Random-variable con- cepts are first introduced for the ,discrete case, where things are particularly simple, and then extended to the continuous case. Chap ter 2 concludes with the topic of derived probability distributions as obtained directly in sample space. Discrete and continuous transform techniques are introduced in Chapter 3. Several applications to sums of independent random varia- bles are included. Contour integration methods for obtaining inverse transforms are not discussed. Chapters 4 and 5 investigate basic random processes involving, respectively, independent and dependent trials. Chapter 4 studies in some detail the Bernoulli and Poisson processes and the resulting families of probability mass and density functions. Because of its significance in experimentation with physi- PREFACE cal systems, the phenomenon of random incidence is introduced in the last section of Chapter 4. Discrete-state Markov models, including both discrete-transi- tion and continuous-transition processes, are presented in Chapter 5. The describing equations and limiting state probabilities are treated, but closed form solutions for transient behavior in the general case are not discussed. Common applications are indicated in the text and in the problems, with most examples based on relatively simple birth- and-death processes. Chapter 6 is concerned .with some of the basic limit theorems, both for the manner in which they relate probabilities to physically observable phenomena and for their use as practical approximations. Only weak statistical convergence is considered in detail. A transform development of the central limit theorem is presented. The final chapter introduces some common issues and techniques of statistics, both classical and Bayesian. My objectives in this obvi- ously incomplete chapter are to indicate the nature of the transition from probability theory to statistical reasoning and to assist the student in developing a critical attitude towards matters of statistical inference. Although many other arrangements are possible, the text is most effectively employed when the chapters are studied in the given order. Examples and Home Problems Many of the


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MIT 6 041 - FUNDAMENTALS OF APPLIED PROBABILITY THEORY

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