Basic probability theory with applications - Lefebvre, Mario (New York : Springer, 2009)
This book presents elementary probability theory with interesting and well-chosen applications that illustrate the theory. An introductory chapter reviews the basic elements of differential calculus which are used in the material to follow. The theory is presented systematically, beginning with the main results in elementary probability theory. This is followed by material on random variables. Random vectors, including the all important central limit theorem, are treated next. The last three chapters concentrate on applications of this theory in the areas of reliability theory, basic queuing models, and time series. Examples are elegantly woven into the text and over 400 exercises reinforce the material and provide students with ample practice. This textbook can be used by undergraduate students in pure and applied sciences such as mathematics, engineering, computer science, finance and economics. A separate solutions manual is available to instructors who adopt the text for their course.
Prefacep. vi List of Tablesp. xiii List of Figuresp. xv Review of differential calculusp. 1 Limits and continuityp. 1 Derivativesp. 3 Integralsp. 7 Particular integration techniquesp. 9 Double integralsp. 12 Infinite seriesp. 14 Geometric seriesp. 15 Exercises for Chapter 1p. 18 Elementary probabilityp. 27 Random experimentsp. 27 Eventsp. 28 Probabilityp. 29 Conditional probabilityp. 32 Total probabilityp. 35 Combinatorial analysisp. 36 Exercises for Chapter 2p. 39 Random variablesp. 55 Introductionp. 55 Discrete casep. 55 Continuous casep. 57 Important discrete random variablesp. 61 Binomial distributionp. 61 Geometric and negative binomial distributionsp. 64 Hypergeometric distributionp. 66 Poisson distribution and processp. 68 Important continuous random variablesp. 70 Normal distributionp. 70 Gamma distributionp. 74 Weibull distributionp. 77 Beta distributionp. 78 Lognormal distributionp. 80 Functions of Random variablesp. 81 Discrete casep. 81 Continuous casep. 82 Characteristics of random variablesp. 83 Exercises for Chapter 3p. 94 Random vectorsp. 115 Discrete random vectorsp. 115 Continuous random vectorsp. 118 Functions of random vectorsp. 124 Discrete casep. 125 Continuous casep. 127 Convolutionsp. 128 Covariance and correlation coefficientp. 131 Limit theoremsp. 135 Exercises for Chapter 4p. 137 Reliabilityp. 161 Basic notionsp. 161 Reliability of systemsp. 170 Systems in seriesp. 170 Systems in parallelp. 172 Other casesp. 176 Paths and cutsp. 178 Exercises for Chapter 5p. 183 Queueingp. 191 Continuous-time Markov chainsp. 191 Quening systems with a single serverp. 197 The M/M/1 modelp. 199 The M/M/1 model with finite capacityp. 207 Queueing systems with two or more serversp. 212 The M/M/s modelp. 212 The M/M/s/c modelp. 218 Exercises for Chapter 6p. 220 Time seriesp. 227 Introductionp. 227 Particular time series modelsp. 235 Autoregressive processesp. 235 Moving average processesp. 244 Autoregressive moving average processesp. 249 Modeling and forecastingp. 251 Exercises for Chapter 7p. 261 List of symbols and abbreviationsp. 269 Statistical tablesp. 275 Solutions to "Solved exercises"p. 281 Answers to even-numbered exercisesp. 325 Answers to multiple choice questionsp. 333 Referencesp. 335 Indexp. 337
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Keywords: central limit theory; queuing models; random variables; random vectors; reliability theory; time series