Fundamentals of Probability, with Stochastic Processes

624 Pages·2018·11.42 MB·English

by  Saeed Ghahramani

Statistics THIRD
EDITION
FUNDAMENTALS OF
PROBABILITY W F FUNDAMENTALS OF
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PROBABILITY
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WITH STOCHASTIC PROCESSES O A
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THIRD EDITION H
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Fundamentals of Probability with Stochastic Processes, Third Edition IC T
teaches probability in a natural way through interesting and instructive exam- A THIRD EDITION
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ples and exercises that motivate the theory, definitions, theorems, and meth- R L
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odology. The author takes a mathematically rigorous approach while closely
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adhering to the historical development of probability. He includes more than E O
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1500 routine and challenging exercises, historical remarks, and discussions
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of probability problems recently published in journals, such as Mathematics E
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Magazine and American Mathematical Monthly. P
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New to the Third Edition
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• Reorganized material to reflect a more natural order of topics
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• 278 new exercises and examples as well as better solutions to the
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SAEED GHAHRAMANI
problems
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• New introductory chapter on stochastic processes
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• More practical, nontrivial applications of probability and stochastic L
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processes in finance, economics, and actuarial sciences, along with more T
genetics examples Y
• New section on survival analysis and hazard functions
• More explanations and clarifying comments in almost every section
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This versatile text is designed for a one- or two-term probability course for
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majors in mathematics, physical sciences, engineering, statistics, actuarial
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science, business and finance, operations research, and computer science. H
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It is also accessible to students who have completed a basic calculus course. A
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K27440
www.crcpress.com
FUNDAMENTALS OF
PROBABILITY
WITH STOCHASTIC PROCESSES
THIRD EDITION
SAEED GHAHRAMANI
Western New England University
Springfield, Massachusetts, USA
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2016 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Version Date: 20151005
International Standard Book Number-13: 978-1-4987-5502-3 (eBook - PDF)
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C
ontents
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Preface xi
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1 Axioms of Probability 1
1.1 Introduction 1
1.2 SampleSpaceandEvents 3
1.3 AxiomsofProbability 10
1.4 BasicTheorems 17
1.5 ContinuityofProbabilityFunction 25
1.6 Probabilities0and1 27
1.7 RandomSelectionofPointsfromIntervals 28
ReviewProblems 33
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2 Combinatorial Methods 36
2.1 Introduction 36
2.2 CountingPrinciple 36
NumberofSubsetsofaSet 40
TreeDiagrams 40
2.3 Permutations 44
2.4 Combinations 50
2.5 Stirling’sFormula 66
ReviewProblems 68
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3 Conditional Probability and Independence 71
3.1 ConditionalProbability 71
ReductionofSampleSpace 75
3.2 LawofMultiplication 80
3.3 LawofTotalProbability 83
3.4 Bayes’Formula 94
3.5 Independence 102
3.6 ApplicationsofProbabilitytoGenetics 119
Hardy-WeinbergLaw 123
Sex-LinkedGenes 125
ReviewProblems 128
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vi Contents
(cid:2) Distribution Functions and
4 131
DiscreteRandom Variables
4.1 RandomVariables 131
4.2 DistributionFunctions 135
4.3 DiscreteRandomVariables 144
4.4 ExpectationsofDiscreteRandomVariables 150
4.5 VariancesandMomentsofDiscreteRandomVariables 165
Moments 170
4.6 StandardizedRandomVariables 173
ReviewProblems 174
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5 Special DiscreteDistributions 177
5.1 BernoulliandBinomialRandomVariables 177
ExpectationsandVariancesofBinomialRandomVariables 183
5.2 PoissonRandomVariable 189
PoissonasanApproximationtoBinomial 189
PoissonProcess 194
5.3 OtherDiscreteRandomVariables 202
GeometricRandomVariable 202
NegativeBinomialRandomVariable 205
HypergeometricRandomVariable 207
ReviewProblems 214
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6 Continuous Random Variables 218
6.1 ProbabilityDensityFunctions 218
6.2 DensityFunctionofaFunctionofaRandomVariable 227
6.3 ExpectationsandVariances 232
ExpectationsofContinuousRandomVariables 232
VariancesofContinuousRandomVariables 238
ReviewProblems 244
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7 Special Continuous Distributions 246
7.1 UniformRandomVariable 246
7.2 NormalRandomVariable 252
CorrectionforContinuity 255
7.3 ExponentialRandomVariables 268
7.4 GammaDistribution 274
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Contents vii
7.5 BetaDistribution 280
7.6 SurvivalAnalysisandHazardFunction 286
ReviewProblems 290
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8 BivariateDistributions 292
8.1 JointDistributionofTwoRandomVariables 292
JointProbabilityMassFunctions 292
JointProbabilityDensityFunctions 296
8.2 IndependentRandomVariables 310
IndependenceofDiscreteRandomVariables 310
IndependenceofContinuousRandomVariables 313
8.3 ConditionalDistributions 322
ConditionalDistributions:DiscreteCase 322
ConditionalDistributions:ContinuousCase 327
8.4 TransformationsofTwoRandomVariables 334
ReviewProblems 342
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9 Multivariate Distributions 346
9.1 JointDistributionofn > 2RandomVariables 346
JointProbabilityMassFunctions 346
JointProbabilityDensityFunctions 354
RandomSample 358
9.2 OrderStatistics 363
9.3 MultinomialDistributions 369
ReviewProblems 373
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10 More Expectations and Variances 375
10.1 ExpectedValuesofSumsofRandomVariables 375
PatternAppearance 382
10.2 Covariance 388
10.3 Correlation 402
10.4 ConditioningonRandomVariables 407
10.5 BivariateNormalDistribution 421
ReviewProblems 425
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viii Contents
(cid:2) Sums of Independent Random
11 428
Variables and Limit Theorems
11.1 Moment-GeneratingFunctions 428
11.2 SumsofIndependentRandomVariables 438
11.3 MarkovandChebyshevInequalities 446
Chebyshev’sInequalityandSampleMean 450
11.4 LawsofLargeNumbers 455
ProportionversusDifferenceinCoinTossing 463
11.5 CentralLimitTheorem 466
ReviewProblems 475
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12 Stochastic Processes 478
12.1 Introduction 478
12.2 MoreonPoissonProcesses 479
WhatIsaQueuingSystem? 490
PASTA:PoissonArrivalsSeeTimeAverage 491
12.3 MarkovChains 494
ClassificationsofStatesofMarkovChains 503
AbsorptionProbability 513
Period 517
Steady-StateProbabilities 518
12.4 Continuous-TimeMarkovChains 530
Steady-StateProbabilities 536
BirthandDeathProcesses 539
12.5 BrownianMotion 548
FirstPassageTimeDistribution 555
TheMaximumofaBrownianMotion 556
TheZerosofBrownianMotion 556
BrownianMotionwithDrift 559
GeometricBrownianMotion 560
ReviewProblems 563
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13 Simulation 568
13.1 Introduction 568
13.2 SimulationofCombinatorialProblems 572
13.3 SimulationofConditionalProbabilities 575
13.4 SimulationofRandomVariables 578
13.5 MonteCarloMethod 586
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Contents ix
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Appendix Tables 590
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Answers to Odd-Numbered Exercises 594
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Index 605
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