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首页麻省理工概率论与数理统计的课程笔记
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LECTURENOTES
Course6.041-6.431
M.I.T.
FALL2000
IntroductiontoProbability
DimitriP.BertsekasandJohnN.Tsitsiklis
ProfessorsofElectricalEngineeringandComputerScience
MassachusettsInstituteofTechnology
Cambridge,Massachusetts
Thesenotesarecopyright-protectedbutmaybefreelydistributedfor
instructionalnonpro�tpruposes.
Contents
1.SampleSpaceandProbability................
1.1.Sets..............................
1.2.ProbabilisticModels.......................
1.3.ConditionalProbability.....................
1.4.Independence..........................
1.5.TotalProbabilityTheoremandBayes’Rule............
1.6.Counting...........................
1.7.SummaryandDiscussion....................
2.DiscreteRandomVariables.................
2.1.BasicConcepts.........................
2.2.ProbabilityMassFunctions...................
2.3.FunctionsofRandomVariables..................
2.4.Expectation,Mean,andVariance.................
2.5.JointPMFsofMultipleRandomVariables.............
2.6.Conditioning..........................
2.7.Independence..........................
2.8.SummaryandDiscussion....................
3.GeneralRandomVariables.................
3.1.ContinuousRandomVariablesandPDFs.............
3.2.CumulativeDistributionFunctions................
3.3.NormalRandomVariables....................
3.4.ConditioningonanEvent....................
3.5.MultipleContinuousRandomVariables..............
3.6.DerivedDistributions......................
3.7.SummaryandDiscussion....................
4.FurtherTopicsonRandomVariablesandExpectations......
4.1.Transforms...........................
4.2.SumsofIndependentRandomVariables-Convolutions.......
iii
ivContents
4.3.ConditionalExpectationasaRandomVariable...........
4.4.SumofaRandomNumberofIndependentRandomVariables....
4.5.CovarianceandCorrelation...................
4.6.LeastSquaresEstimation....................
4.7.TheBivariateNormalDistribution................
5.TheBernoulliandPoissonProcesses..............
5.1.TheBernoulliProcess......................
5.2.ThePoissonProcess.......................
6.MarkovChains.......................
6.1.Discrete-TimeMarkovChains..................
6.2.Classi�cationofStates......................
6.3.Steady-StateBehavior......................
6.4.AbsorptionProbabilitiesandExpectedTimetoAbsorption.....
6.5.MoreGeneralMarkovChains...................
7.LimitTheorems.......................
7.1.SomeUsefulInequalities.....................
7.2.TheWeakLawofLargeNumbers.................
7.3.ConvergenceinProbability....................
7.4.TheCentralLimitTheorem...................
7.5.TheStrongLawofLargeNumbers................
Preface
Theseclassnotesarethecurrentlyusedtextbookfor“ProbabilisticSystems
Analysis,”anintroductoryprobabilitycourseattheMassachusettsInstituteof
Technology.Thetextofthenotesisquitepolishedandcomplete,buttheprob-
lemsarelessso.
Thecourseisattendedbyalargenumberofundergraduateandgraduate
studentswithdiversebackgrounds.Acccordingly,wehavetriedtostrikeabal-
ancebetweensimplicityinexpositionandsophisticationinanalyticalreasoning.
Someofthemoremathematicallyrigorousanalysishasbeenjustsketchedor
intuitivelyexplainedinthetext,sothatcomplexproofsdonotstandintheway
ofanotherwisesimpleexposition.Atthesametime,someofthisanalysisand
thenecessarymathematicalresultsaredeveloped(atthelevelofadvancedcalcu-
lus)intheoreticalproblems,whichareincludedattheendofthecorresponding
chapter.Thetheoreticalproblems(markedby*)constituteanimportantcom-
ponentofthetext,andensurethatthemathematicallyorientedreaderwill�nd
hereasmoothdevelopmentwithoutmajorgaps.
Wegivesolutionstoalltheproblems,aimingtoenhancetheutilityof
thenotesforself-study.Wehaveadditionalproblems,suitableforhomework
assignment(withsolutions),whichwemakeavailabletoinstructors.
Ourintentistograduallyimproveandeventuallypublishthenotesasa
textbook,andyourcommentswillbeappreciated
DimitriP.Bertsekas
bertsekas@lids.mit.edu
JohnN.Tsitsiklis
jnt@mit.edu
v
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