應(yīng)用隨機(jī)過程-教學(xué)大綱

《應(yīng)用隨機(jī)過程》教學(xué)大綱“AppliedStochasticProcess”CourseOutline課程編號：152063A課程類型：專業(yè)選修課總學(xué)時：48講課學(xué)時：48實(shí)驗(yàn)（上機(jī)）學(xué)時：0學(xué)分：3適用對象：經(jīng)濟(jì)學(xué)、管理學(xué)、統(tǒng)計(jì)學(xué)、金融學(xué)等先修課程：概率論與數(shù)理統(tǒng)計(jì)、線性代數(shù)、微積分CourseCode:152063ACourseType:DisciplinebasiccoursePeriods:48Lecture:48Experiment(Computer):0Credits:3ApplicableSubjects:Economics,Management,Statistics,Financeetc.PreparatoryCourses:ProbabilityandMathematicalStatistics,LinearAlgebra,MathematicalAnalysis一、課程的教學(xué)目標(biāo)這是一門向經(jīng)濟(jì)學(xué)和管理學(xué)相關(guān)專業(yè)本科生介紹隨機(jī)過程的理論方法和實(shí)際應(yīng)用的專業(yè)選修課程。本課程在學(xué)生已經(jīng)扎實(shí)掌握概率論和數(shù)理統(tǒng)計(jì)基礎(chǔ)知識的前提下，介紹隨機(jī)過程中的基本概念和結(jié)果。本課程主要訓(xùn)練學(xué)生的如下能力：（1）靈活組合運(yùn)用微積分，線性代數(shù)和概率論解決數(shù)學(xué)問題的能力；（2）進(jìn)一步的抽象思維和符號運(yùn)算能力；（3）把實(shí)際問題抽象為理論模型，再把理論結(jié)果結(jié)合實(shí)際情況進(jìn)行解釋的能力；（4）利用計(jì)算機(jī)和MATLAB軟件解決復(fù)雜計(jì)算問題和無解析解的問題的能力。學(xué)習(xí)完本課程后，學(xué)生們能對隨機(jī)過程及其應(yīng)用有基本的認(rèn)識，并且具有今后進(jìn)一步學(xué)習(xí)高級隨機(jī)過程理論,現(xiàn)代金融工程和隨機(jī)控制理論和從事相關(guān)工作的專業(yè)基礎(chǔ)。ThecourseofAppliedStochasticProcessintroducestheoryandapplicationofstochasticprocesstoundergraduatestudents.Studentsareassumedtohavealreadyfinishedtheirstudyofundergraduatelevelprobabilityandstatistics.Studentstrainthefollowingabilitiesthiscourse:(1)methodologicallyapplyingcalculus,linearalgebraandprobabilitytheorytonewmathematicalproblems;(2)advancedlogicalreasoningandsymbolhandling;(3)buildingmathematicalmodelsfromrealworldproblems,andthentranslatingmathematicalresultsbacktofittheoriginalquestion;(4)employingcomputersandMATLABsoftwaretosolvecomputationallycomplexedproblemsand/orproblemswithoutclosedformsolution.Uponfinishingthecourse,studentscangainabasicunderstandingofthetheoryandapplicationofstochasticprocess,andbuildafoundationforstudyingadvancedstochasticprocesstheory,modernfinancialengineeringandstochasticcontroltheory,aswellasperformingrelevantwork.二、教學(xué)基本要求本課程講述隨機(jī)過程的基礎(chǔ)理論結(jié)果及其應(yīng)用。隨機(jī)過程是比隨機(jī)變量更深入一步的數(shù)學(xué)概念，理解隨機(jī)過程必須建立在學(xué)生扎實(shí)深入的學(xué)好概率論，掌握隨機(jī)變量的性質(zhì)的基礎(chǔ)上。同時，本課程還要求學(xué)生對微積分和線性代數(shù)有深刻的掌握，并能靈活運(yùn)用已有知識去解決新問題。隨機(jī)過程屬于數(shù)學(xué)中較難理解的概念，同時學(xué)生們很可能在先修課程中就有不明白的地方。因此，任課教師需要精心設(shè)計(jì)教學(xué)計(jì)劃和編寫課件，做到能深入淺出的闡述相關(guān)理論。同時在授課中要及時與學(xué)生交流并發(fā)現(xiàn)學(xué)生知識結(jié)構(gòu)中的薄弱點(diǎn)有針對性的進(jìn)行講解。本課程重點(diǎn)講解隨機(jī)過程中的最基本要素：泊松過程，更新過程和馬爾科夫過程。盡量做到讓學(xué)生對這些最重要的概念有全面深刻的了解，從而能舉一反三學(xué)習(xí)更廣泛深入的知識。本課程將講述大量例子來與理論相結(jié)合。并計(jì)劃安排少量計(jì)算機(jī)實(shí)踐來向?qū)W生解釋實(shí)際中處理問題的方法。教學(xué)中，鼓勵學(xué)生課前預(yù)習(xí)，課上安排課堂討論，提高學(xué)生的課堂參與積極性，以便學(xué)生能夠深入理解知識要點(diǎn)；課后讓學(xué)生完成作業(yè)并對部分習(xí)題進(jìn)行課堂講解。課程的考核方式及其所占權(quán)重如下：出勤10%作業(yè)20%期末閉卷考試70% 在上述考核方式中，作業(yè)來自于課本中的理論和實(shí)踐習(xí)題；期末閉卷考試考查理論知識和解決應(yīng)用題的能力。Thiscourseintroducesbasicresultsandapplicationsofstochasticprocess.Stochasticprocesslaysitsfoundationonandisafurtherdevelopmentoftheconceptofrandomvariables.Forthisreason,studentsareexpectedtohavealreadyhadafirmgraspofundergraduatelevelprobabilityandstatisticknowledge,aswellascalculusandlinearalgebra.Studentsshouldalsobeabletousewhattheyalreadyhaveknownmethodologicallytosolvenewproblems.Stochasticprocessfallsintothecategoryof“hard”mathematicsforundergraduatelevelstudents.Ontheotherhand,mostundergraduatestudentshavedifficultiesunderstandingatleastcertainmathematicalknowledgetheyhavealreadylearned.Underthesecircumstances,thelecturerisrequiredtocarefullydesignthestructureofinstructionandthelecturenotestoensurehardconceptsareinstilledstepbystepfollowingspecialexamplesandconceptseasiertounderstand.Duringtheinstruction,thelecturershouldbekeentospotweakpointsofstudents’knowledgestructureandcleartheobstaclehinderingprogress.Thecourseisbuiltaroundthreefundamentalconceptsinstochasticprocess:Poissonprocess,renewalprocessandMarkovprocess.Oncestudentsareabletocomprehendthoseseminalbuildingblocks,theywillexploreothervastareasofthissubjectmuchfaster.Thecoursewillintroducenumerousexamplesalongwiththeoryknowledge.Studentsareencouragedtoreadrelevantmaterialsbeforetheclass.Discussionswillbeorganizedintheclasstofacilitatebetterunderstanding.Problemsetsisgivenafterclass,andsomeofthemwillbeanalyzedinlaterclasses.Themethodsofevaluationofthiscourseandtheirweightsareasfollows:Attendance10%Assignments20%FinalExam(closed)70% Inthemethodsofevaluationabove,theassignmentscomefromthetheoreticalandapplicationexercisesinthetextbook.Theclosedformfinalexamfocusesontheexaminationofknowledgeintheory.三、各教學(xué)環(huán)節(jié)學(xué)時分配教學(xué)課時分配序號章節(jié)內(nèi)容講課實(shí)驗(yàn)其他合計(jì)緒論Introduction11概率論回顧ReviewofProbabilityTheory22泊松過程PoissonProcess415非齊次泊松過程N(yùn)onhomogeneousPoissonProcess213復(fù)合泊松過程CompoundPoissonProcess22排隊(duì)論基礎(chǔ)ElementaryQueuingTheory314泊松到達(dá)平均PoissonArrivalSeeTimeAverage11數(shù)列的生成函數(shù)ThegeneratingfunctionofaSequence11更新過程RenewalProcess819剩余壽命，當(dāng)前壽命和總觀測壽命ExcessLife,CurrentLifeandTotalLife617離散馬爾科夫鏈Discrete-timeMarkovChains617連續(xù)馬爾科夫鏈Continuous-timeMarkovChains426合計(jì)40848四、教學(xué)內(nèi)容緒論什么是隨機(jī)過程第一個例子：股票價(jià)格連續(xù)和離散隨機(jī)變量分布函數(shù)和密度函數(shù)期望和方差聯(lián)合分布和獨(dú)立教學(xué)重點(diǎn)、難點(diǎn)：隨機(jī)過程概念課程的考核要求：無Chapter1IntroductionSection1Whatisastochasticprocess?Section2Firstexample:thedynamicofstockpriceSection3DiscreteandcontinuousrandomvariableSection4ProbabilityandcumulativedensityfunctionsSection5ExpectationandvarianceofrandomvariablesSection6JointdistributionandindependenceKeyandDifficultPoints:theconceptionofastochasticprocessEvaluationRequirements:no第二章概率論回顧第一節(jié)重要的離散分布伯努利分布二項(xiàng)分布泊松分布幾何分布負(fù)二項(xiàng)分布第二節(jié)重要的連續(xù)分布2.1均勻分布2.2正態(tài)分布2.3對數(shù)正態(tài)分布第三節(jié)矩和矩生成函數(shù)第四節(jié)馬爾科夫不等式第五節(jié)大數(shù)定律第六節(jié)中心極限定理第七節(jié)隨機(jī)變量的轉(zhuǎn)換7.1分布函數(shù)法7.2密度函數(shù)法7.3多元隨機(jī)變量的轉(zhuǎn)換7.4示例：對數(shù)正態(tài)分布的密度函數(shù)第八節(jié)貝葉斯公式第九節(jié)條件期望，條件矩，條件方差第十節(jié)拉普拉斯變換第十一節(jié)卷積教學(xué)重點(diǎn)、難點(diǎn)：條件期望和隨機(jī)變量的轉(zhuǎn)換課程的考核要求：掌握現(xiàn)代概率論基礎(chǔ)中的基本概念和定理Chapter2ReviewofprobabilitytheorySection1Importantdiscretedistributions1.1Bernoullidistribution1.2Binominaldistribution1.3Poissondistribution1.4Geometricdistribution1.5NegativebinominaldistributionSection2Importantcontinuousdistributions2.1Uniformdistribution2.2Normaldistribution2.3Log-normaldistributionSection3MomentandmomentgeneratingfunctionSection4MarkovinequalitySection5LawoflargenumbersSection6CentralLimitingTheoremSection7Transformationofrandomvariables7.1CDFmethod7.2PDFmethod7.3Transformationofmulti-dimensionalrandomvariables7.4Examples:thePDFoflog-normaldistributionSection8TheBayesianformulaSection9Conditionalexpectation,conditionalmomentsandconditionalvarianceSection10LaplacetransformsSection11ConvolutionKeyandDifficultPoints:conditionalexpectationandtransformationofrandomvariablesEvaluationRequirements:understandfundamentalsofmodernprobabilitytheory,includingnotationsandtheorems第三章泊松過程第一節(jié)泊松過程的定義獨(dú)立增量泊松過程在給定時間點(diǎn)呈現(xiàn)泊松分布示例：電話服務(wù)中心第二節(jié)泊松過程的基本假設(shè)2.1計(jì)數(shù)過程2.2連續(xù)時間域上的離散過程2.3平穩(wěn)獨(dú)立增量2.4多個到達(dá)的概率為高階無窮小第三節(jié)泊松過程的分布公式第四節(jié)到達(dá)時間和間隔到達(dá)時間第五節(jié)泊松過程，到達(dá)時間和間隔到達(dá)時間之間的等價(jià)關(guān)系第六節(jié)生成泊松到達(dá)時間第七節(jié)示例：有線電視公司教學(xué)重點(diǎn)、難點(diǎn)：泊松過程的基本性質(zhì)以及泊松過程和到達(dá)時間及間隔到達(dá)時間的等價(jià)課程的考核要求：了解泊松過程的定義和基本性質(zhì)Chapter3PoissonProcessSection1DefinitionofPoissonprocessIndependentincrementPoissonprocessatagiventimeisaPoissondistributionAnexampleofaphonecallcenterSection2GeneralassumptionsofPoissonprocess2.1Countingprocess2.2Discretevaluedonacontinuoustimedomain2.3Stationaryandindependentincrement2.4HigherorderinfinitesimalformultiplearrivalsSection3ThedistributionformulaofthePoissonprocessSection4ArrivaltimeandinterarrivaltimeSection5EquivalencebetweenthePoissonprocess,thearrivaltimesandtheinterarrivaltimesSection6GeneratingPoissonarrivaltimesSection7AnexampleofcablecompanyKeyandDifficultPoints:fundamentalpropertiesofPoissonprocessandtheequivalencebetweenitanditsarrivaltimesandinterarrivaltimesEvaluationRequirements:understandthedefinitionofPoissonprocessanditsbasicproperties第四章非齊次泊松過程第一節(jié)定義變化的到達(dá)率獨(dú)立增量多個到達(dá)概率為高階無窮小第二節(jié)非齊次泊松過程的間隔到達(dá)時間和到達(dá)時間以及非齊次泊松過程的分布2.1非齊次泊松過程和間隔到達(dá)時間和到達(dá)時間的等價(jià)2.2首次到達(dá)時間的分布2.3后續(xù)間隔到達(dá)時間的條件分布2.4在給定時間內(nèi)出現(xiàn)n次到達(dá)的概率2.5非齊次泊松過程分布的推導(dǎo)第三節(jié)非齊次泊松過程是特殊的齊次泊松過程第四節(jié)各種非齊次泊松過程之間的轉(zhuǎn)換第五節(jié)非齊次泊松過程和齊次泊松過程之間的轉(zhuǎn)換第六節(jié)示例：出租車服務(wù)第七節(jié)非齊次泊松過程的應(yīng)用教學(xué)重點(diǎn)、難點(diǎn)：非齊次泊松過程和齊次泊松過程的聯(lián)系和區(qū)別課程的考核要求：掌握非齊次泊松過程的性質(zhì)和分布Chapter4NonhomogeneousPoissonprocessSection1DefinitionTime-varyingarrivalrateIndependentincrementHigherorderinfinitesimalformultiplearrivalsSection2InterarrivaltimesandarrivaltimesofnonhomogeneousPoissonprocessanddistributionofnonhomogeneousPoissonprocess2.1EquivalencebetweennonhomogeneousPoissonprocessanditsinterarrivaltimesandarrivaltimes2.2Distributionofthefirstinterarrivaltime2.3Conditionaldistributionofconsequentialinterarrivaltimes2.4Theprobabilityofaspecificarrivalcasewithnarrivalsonagiventimeinterval2.5GettingthedistributionofnonhomogeneousPoissonprocessSection3NonhomogeneousPoissonprocessasaspecialcaseofPoissonprocessSection4TransformationbetweendifferentnonhomogeneousPoissonprocessesSection5FromnonhomogeneoustohomogeneousPoissonprocessSection6Example:CarserviceonGalvestonBeachSection7ApplicationsofnonhomogeneousPoissonprocessKeyandDifficultPoints:theresemblanceanddistinctionbetweennonhomogeneousandhomogeneousPoissonprocessesEvaluationRequirements:understanddefinition,propertyanddistributionofnonhomogeneousPoissonprocess第五章復(fù)合泊松過程第一節(jié)定義第二節(jié)復(fù)合泊松過程的性質(zhì)2.1復(fù)合泊松過程的獨(dú)立平穩(wěn)增量2.2復(fù)合泊松過程的方差大于期望2.3復(fù)合泊松過程的期望分解第三節(jié)復(fù)合泊松過程的推廣第四節(jié)“口吃”過程及其應(yīng)用第五節(jié)對數(shù)泊松過程教學(xué)重點(diǎn)、難點(diǎn)：復(fù)合泊松過程和構(gòu)成它的泊松過程的聯(lián)系課程的考核要求：掌握復(fù)合泊松過程的構(gòu)造以及它的幾個特殊性質(zhì)Chapter5CompoundPoissonprocessSection1DefinitionSection2Properties2.1Independenceandincrement-stationarityofcompoundPoissonprocessfromtheunderlyingPoissonprocess2.2Varianceisalwaysgreaterthanexpectation2.3DecompositionofexpectationSection3GeneralizationofcompoundPoissonprocessSection4ThestutteringPoissonprocessanditsapplicationSection5ThelogarithmicPoissonprocessKeyandDifficultPoints:thepropertyofcompoundPoissonprocesscomesfromitsunderlyingPoissonprocessEvaluationRequirements:understandthedefinitionandseveralspecialpropertyofcompoundPoissonprocess第六章排隊(duì)論基礎(chǔ)第一節(jié)排隊(duì)論是現(xiàn)實(shí)世界中隊(duì)列的抽象第二節(jié)隨機(jī)隊(duì)列的構(gòu)造2.1顧客的到達(dá)過程2.2服務(wù)器和服務(wù)時間的分布2.3等待空間第三節(jié)無限個服務(wù)器的隊(duì)列第四節(jié)有限個服務(wù)器的隊(duì)列4.1等待服務(wù)和隊(duì)列的形成4.2服務(wù)時間的指數(shù)分布和多個服務(wù)器的并行服務(wù)速率第五節(jié)狀態(tài)概率向量第六節(jié)隊(duì)列系統(tǒng)的微分差分方程6.1狀態(tài)之間的轉(zhuǎn)換6.2當(dāng)存在空閑服務(wù)器時6.3當(dāng)不存在空閑服務(wù)器時6.4轉(zhuǎn)移向量和轉(zhuǎn)移矩陣第七節(jié)有限等待空間第八節(jié)“口吃”泊松隊(duì)列第九節(jié)兩個隨機(jī)隊(duì)列的級聯(lián)教學(xué)重點(diǎn)、難點(diǎn)：隨即隊(duì)列的狀態(tài)空間和轉(zhuǎn)移矩陣課程的考核要求：了解隨機(jī)隊(duì)列和現(xiàn)實(shí)隊(duì)列的聯(lián)系以及隨機(jī)隊(duì)列的數(shù)學(xué)描述。Chapter6ElementaryqueuingtheorySection1QueuingtheoryasageneralizationfromtheactualqueueandwaitingpracticeintherealworldSection2Structureofastochasticqueue2.1Thearrivalprocessofcustomers2.2Theserversanditsassociatedservicetimedistribution2.3ThewaitingspaceSection3ThequeuewithinfinitenumberofserversSection4Thequeuewithfinitenumberofservers4.1Waitingforserviceandformationofthequeue4.2ExponentialdistributionassumptionofservicetimeandservicerateofmultipleserversSection5StateprobabilityvectorofthequeueSection6Thedifferential-differenceequationsofthequeuesystem6.1Transitionbetweenstates6.2Thecasesofnotallserversareoccupied6.3Whentherearemorecustomersthanservers6.4Thetransitionvectors6.5ThetransitionmatrixSection7LimitedwaitingspaceSection8ThestutteringPoissonprocessSection9TwoqueuesystemsintandemKeyandDifficultPoints:ThestatespaceandthetransitionmatrixofstochasticqueueEvaluationRequirements:understandrelationshipbetweenstochasticqueueandrealworldqueue,andthemathematicaldescriptionofthestochasticqueue第七章泊松到達(dá)時間平均(PASTA)第一節(jié)隨即隊(duì)列系統(tǒng)的漸進(jìn)分布和穩(wěn)態(tài)第二節(jié)隨機(jī)隊(duì)列的時間平均和顧客平均第三節(jié)缺乏預(yù)期假設(shè)和到達(dá)定理第四節(jié)指數(shù)服務(wù)時間隊(duì)列的穩(wěn)態(tài)分布教學(xué)重點(diǎn)、難點(diǎn)：漸進(jìn)時間平均和顧客平均等效的條件課程的考核要求：了解泊松到達(dá)時間的概念和它的重要性，掌握指數(shù)服務(wù)時間隊(duì)列達(dá)到穩(wěn)態(tài)的條件和穩(wěn)態(tài)下的狀態(tài)分布公式Chapter7PoissonArrivalSeeTimeAverage(PASTA)Section1Thelong-timedistributionofthequeuesystemandthesteadystateSection2ThetimeaverageandthecustomeraverageofthequeuesystemSection3LackofAnticipationAssumption(LAA)andthearrivaltheoremSection4SteadystatevectorofthequeuesystemwithexponentialservicetimeKeyandDifficultPoints:understandwhentimeaveragewillbeequaltocustomeraverageEvaluationRequirements:understandPASTAanditsimportance,andtheconditionforanexponentialqueuetoreachsteadystateanditsstateprobabilitydistributionunderthesteadystate第八章數(shù)列的生成函數(shù)第一節(jié)數(shù)列和它的生成函數(shù)第二節(jié)數(shù)列生成函數(shù)的性質(zhì)第三節(jié)常見數(shù)列的生成函數(shù)第四節(jié)離散隨機(jī)變量的生成函數(shù)教學(xué)重點(diǎn)、難點(diǎn)：生成函數(shù)之間的關(guān)系取決于數(shù)列之間的關(guān)系課程的考核要求：了解數(shù)列的生成函數(shù)的概念和基本性質(zhì)，會求解簡單數(shù)列的生成函數(shù)和離散隨機(jī)變量的生成函數(shù)Chapter8ThegeneratingfunctionofasequenceSection1ThecorrespondencebetweenasequenceandacompositepowerseriesSection2PropertiesofthegeneratingfunctionofasequenceSection3GeneratingfunctionsofcommonsequencesSection4ProbabilitygeneratingfunctionofadiscreterandomvariableKeyandDifficultPoints:therelationshipofgeneratingfunctionsdependsontherelationshipbetweentheoriginalsequencesEvaluationRequirements:understandthedefinitionandbasicpropertyofgeneratingfunctionofsequence,solveforgeneratingfunctionsofsimplesequencesanddiscreterandomvariables.第九章更新過程更新過程是泊松過程的推廣更新過程和它的到達(dá)時間和間隔到達(dá)時間之間的等價(jià)關(guān)系更新過程的分布更新函數(shù)和更新密度更新密度的終值更新方程和更新方程的解根本更新定理停時及其應(yīng)用，Wald定理教學(xué)重點(diǎn)、難點(diǎn)：掌握得到更新方程首次到達(dá)時間期望法課程的考核要求：了解更新方程的概念，了解更新函數(shù)，更新密度，更新方程和它的解，了解停時的概念和性質(zhì)Chapter9TherenewalprocessSection1RenewalprocessasageneralizationofPoissonprocessSection2EquivalencebetweenrenewalprocessanditsarrivalandinterarrivaltimesSection3DistributionoftherenewalprocessSection4TherenewalfunctionandrenewaldensitySection5ThefinalvalueoftherenewaldensitySection6Therenewal-typeequationsandtherenewal-typesolutionSection7ThekeyrenewaltheoremSection8Stoppingtimesanditsapplications,Wald’stheoremKeyandDifficultPoints:Conditioningonfirstarrivaltimetoobtaintherenewal-typeequationEvaluationRequirements:understandthedefinitionofrenewalprocess,therenewalfunctionandrenewaldensity,knowhowtherenewal-typeequationisformedandhowitissolved,understandstoppingtimesanditspropertiesandapplications.第十章剩余壽命，當(dāng)前壽命與總觀測壽命更新過程的生滅性質(zhì)和當(dāng)前對象總觀測壽命和間隔到達(dá)時間的區(qū)別剩余壽命分布的更新方程及其解剩余壽命密度和它的終值當(dāng)前壽命與剩余壽命的等價(jià)關(guān)系和當(dāng)前壽命的分布總觀測壽命的分布間隔到達(dá)時間為均勻分布的更新過程生滅過程離散更新過程教學(xué)重點(diǎn)、難點(diǎn)：對三種壽命的理解，總觀測壽命和間隔到達(dá)時間的區(qū)別，壽命之間的等價(jià)關(guān)系課程的考核要求：了解三種壽命的定義和它們之間的聯(lián)系，了解總觀測壽命和間隔到達(dá)時間的區(qū)別，能寫出三種壽命的更新方程，能解釋三種壽命及其密度分布的終值定理的意義Chapter10ExcessLife,CurrentLifeandTotalLifeSection1TherenewalprocessasabirthanddeathprocessandthecurrentoccupantatagiventimeSection2DifferencebetweenthetotallifeandtheinterarrivaltimeSection3Renewal-typeequationforthedistributionofexcesslifeanditssolutionSection4ExcesslifedensityanditsfinalvalueSection5TheequivalencebetweenthecurrentlifeandtheexcesslifeandthedistributionofthecurrentlifeSection6ThedistributionofthetotallifeSection7ArenewalprocesswithuniforminterarrivaltimesSection8RegenerativeprocessSection9DiscreterenewalprocessKeyandDifficultPoints:thedefinitionandbehaviorofthreelives,thedistinctionbetweentotallifeandinterarrivaltimeEvaluationRequirements:understandthedefinitionandrelationshipbetweenexcesslife,currentlifeandtotallife,understandthedifferencebetweentotallifeandinterarrivaltime,beabletowritedowntherenewal-typeequationofthreelives,beabletoexplaintheintuitionbehindthedistributionofth