Transcription of SC505 STOCHASTIC PROCESSES Class Notes
1 SC505 STOCHASTICPROCESSESC lassNotesc Casta~non & Electricaland ComputerEngineeringBostonUniversityColle geof Engineering8 St. Mary'sStreetBoston,MA02215 Fall 20042 Contents1 Introductionto Probability .. andIndependenceof Events.. RandomVariables.. RandomVariables.. RandomVariables.. ,Densities,andExpectations.. theCovarianceMatrix.. inequality.. Inequality .. 'sInequality .. Inequalities..372 Sequencesof .. of LargeNumbers.. Convergence.. theLaw of LargeNumbersandtheCentralLimitTheorem.
2 RandomVariables..523 .. StochasticProcesses.. of StochasticProcesses.. StochasticProcesses.. StochasticProcesses.. RandomProcesses.. :Phase-ShiftKeying.. Functionsof VectorProcesses.. of Wide-senseStationaryProcesses.. SpectralDensity of Wide-SenseStationaryProcesses..714 of StochasticProcesses.. erentiation.. erentiationof GaussianStochasticProcesses.. of StationaryRandomProcesses..865 .. Continuous-timeLinearSystems.. Discrete-timeLinearSystems.. MultivariableSystems..996 Samplingof.
3 1057 ModelIdenti Models.. MovingAverage(ARMA)Models.. 1168 .. RiskApproach andtheLikelihood RatioTest.. OperatingCharacteristic.. theROC.. 1469 SeriesExpansionsandDetectionof .. StochasticProcesses.. KnownSignalsin Additive WhiteNoise.. UnknownSignalsin WhiteNoise.. KnownSignalsin ColoredNoise.. 157 CONTENTS510 Estimationof .. DecisionRule.. DecisionRulePerformance.. LeastSquareEstimation.. MaximumA Posteriori(MAP)Estimation.. LinearLeastSquare(LLSE)Estimation.
4 Estimation.. MAPestimation.. 18711 LLSEE stimationof .. :TheWiener-HopfEquation.. (WienerSmoothing).. 20912 Recursive .. Estimationof a RandomVector.. UpdateStep.. theWienerandKalmanFilter.. 22113 DiscreteStateMarkov ,DiscreteValuedMarkov PROCESSES .. ,DiscreteValuedMarkov PROCESSES .. PoissonProcesses.. 233A UsefulTransforms235B .. 242C Summaryof .. Transformation.. niteMatrices.. 254D Thenon-zeromeancase257 Listof Times k.. timesT(n) andinterarrival times k.
5 (PCP)N(t) andtherelationshipbetweenarrival timesT(n)andinterarrival times k.. a deterministicdecisionruleas a divisionof theobservationspaceinto disjointregions,illustratedhereforthecas eof two .. we obtainmoreindependent .. discretevaluedproblemof .. discretevaluedproblemof .. theperformanceof a .. theoverallROCobtainedfora .. :Illustrationof theexpectedcostof a decisionruleusinganarbitrary xedthresholdasa functionof thetruepriorprobabilityP 1. Themaximumcostof thisdecisionruleis at theleftendpoint.
6 Thelower curve is thecorrespondingexpectedcostof theoptimalLRT. Right:Theexpectedcostof theminimaxdecisionruleas a functionof thetruepriorprobabilityP 1. by intersecting( ) fora .. ,decisionregionsandPFfortheproblemof .. thespaceof thelikelihoods .. thedecisionrulein .. thedecisionrulein thelikelihood .. theMLdecisionrulein .. decisionrulein .. thecalculationof Pr (DecideH0jH1) in .. thecalculationof Pr (DecideH1jH1) in .. theMLEstimator:(a)pYjX(yjx) viewed as a functionofyfor xedvaluesofx, (b)pYjX(yjx) viewed as a functionofxfor xedy, (c)pYjX(yjx) viewed as a functionof bothxandy.
7 For a givenobservationy0,bxM L(y) is themaximumwithrespecttoxforthegiveny=y0. . BasedonRelative Timesof .. noncausalWiener lterof .. spectraof .. to .. (t), thepole-zeroplot, .. (t), thepole-zeroplot, .. (t), thepole-zeroplot, .. (t), thepole-zeroplot, .. (t) forT >0 andT <0.. (s).. pole-zerosymmetrypropertiesofSY Y(s).. WienerFilterSolutions.. :Estimate.. :CovarianceandGain.. 229 Listof probability distributionfunctionandprobability density .. randomvariables.
8 (N/AunderthePDFcolumnindicatesthattherei s no simpli edform.).. MAPandMLEstimationfora .. thecausalWiener lterandtheKalman .. 23910 LISTOFTABLESC hapter1 Introductionto ProbabilityWhatis probability theory?It is anaxiomatictheorywhichdescribes andpredictstheoutcomesof inexact, theabove de probabilisticanalysisis todetermineorestimatetheprobabilitiestha tcertainknownevents occur,andthentousetheaxiomsofprobability theoryto combinethisinformationto derive probabilitiesof otherevents of interest,andtopredicttheoutcomesof example,considerany is theshu ingof a deck of cards,withtheoutcomebeingtheorderin which theunderlyingprobabilitieswouldbe thatallorderingsareequallylikely.
9 Theunderlyingevents wouldthenbe assigneda of theevents,youmay wishto computetheprobability that,if youareplayingaloneagainsta dealer,youwouldwina handof thecardsleadto winninghands,andtheprobability of winningcanbe whatwe meanby theprobability of anevent that,if anexperiment is repeatedanin nitenumber of times,thefractionofexperiments in which theevent occursis itsprobability. Ontheotherhand,thesubjectivistinterpreta tionisthata probability represents anindividualbeliefthata certainevent mostappropriatewhenexperiments cannotbe repeated,such as in which interpretationis used,thesameaxiomatictheoryis thischapter,we reviewsomeof thekey backgroundconceptsin probability ProbabilityA probability spaceis a triple( ;F; P) which is usedto describe theoutcomesof a is thesetof allpossibleelementaryexperiment outcomes!
10 ThesetFis a collectionof subsetsof which is ,ifAi2 F; i= 1; : : :, thenAi ;[1i=1Ai2F; Ai2F. (Thenotation Ais usedto denotethecomplement ofA, or A= A, whereB A=fxjx2Bx62Ag.)ThesetFis calleda - eldbecauseof itsclosureundercountableunion,andis referredto as thesetof Fis calledanevent. Notethattheabove propertiesalsoimplythatFis probability valuein [0;1] to each event containedinF; thatis,it mapsthesetof events into theclosedunitinterval [0;1].Furthermore,theprobability measurehassomeimportantproperties, eventsA,Baresaidto be mutuallyexclusive ifA\B=;, theempty aprobability ( )= (A) 0 ([1i=1Ai) =P1i=1P(Ai) ifAi\Aj=;foralli6=j.]]
