概率统计第2章课件

2Randomvariablesandtheirdistributions2.1RandomVariables2.2Discreterandomvariables2.3SomeImportantDiscreteProbabilityDistributions2.5ContinuousRandomVariables2.6SomeContinuousProbabilityDistributions2.7FunctionsofRandomVariables2.4CumulativeDistributionFunctions2Randomvariablesandtheird12.1RandomVariablesExample1Tosscoin:1HT2.1RandomVariablesExample12RSExample2Testthelifeinyearsoflightbulbs:Definition2.1LetSbethesamplespaceassociatedwithaparticularexperiment.Asingle-valuedfunctionXassigningtoeveryelementarealnumber,X(ω),iscalleda

randomvariable.DenotedbyX.2RSExample2Testthelifein3Ingeneral,Definition2.1LetSbethesamplespaceassociatedwithaparticularexperiment.Asingle-valuedfunctionXassigningtoeveryelementarealnumber,X(ω),iscalledarandomvariable.DenotedbyX.andx,y,z…representarealnumber.weuseX,Y,Z….representarandomvariableNoticethatRXisalwaysasetofrealnumbers.Definition2.2

ThesetofallpossiblevaluesofXiscalledtherangespaceofX

andisdenotedbyRX.3Ingeneral,Definition2.1Let4Definition2.2

ThesetofallpossiblevaluesofXiscalledtherangespaceofX

andisdenotedbyRX.NoticethatRXisalwaysasetofrealnumbers.Foraboveexample,4Definition2.2Foraboveexamp5Foraboveexample,5Foraboveexample,56

Randomvariablecouldtakedifferentvaluesdependingondifferentrandomexperiments.Becausetheexperimentresultsshowuprandomlytherandomvariablecouldtakevaluesdependingoncertainkindofprobability.(2)Thewaytotakethevaluesforrandomvariableobeyskindofprobabilityrule.

Randomvariableiskindoffunction,butitisessentiallydifferenttotheothergeneralfunctions.Thelaterkindoffunctionsaredefinedonrealnumbersetwhilerandomvariablesaredefinedonthesamplespacewhoseelementswouldnotallberealnumbers.2.Notes(1)Randomvariableisdifferenttothecommonfunction6Randomvariablecouldtaked7

Theconceptofrandomeventiscontainedwithintheconceptofrandomvariable,whichismoreextended.Fromanotherpointofviewwecansayrandomeventistosearchtherandomphenomenabyastaticmethodwhilerandomvariableistodosobyadynamicway.(3)Therelationshipbetweenrandomevent&variable7Theconceptofrandomevent8CategoriesofrandomvariableDiscreteObservethenumberdisplayedonarollingdice.PossiblevaluesforarandomvariableX:RandomVariableContinuouse.g.11,2,3,4,5,6.Non-discreteOthers(1)Discrete

ifthenumberofvaluesarandomvariablecouldtakeisfiniteorcountableinfinitethenthisvariableiscalleddiscreterandomvariable.8CategoriesofrandomvariableD99e.g.2LetXbearandomvariablerepresenting“Thenumberofshootingsasoneshootscontinuouslyuntilthetargetisshot.”.ThenthepossiblevaluesXcouldtake:e.g.3

Iftheprobabilityforoneshootertoshootthetargetis0.8,nowhehasshot30timesandletXbetherandomvariabletorepresent“Thenumberofshootingsthatareshotonthetarget”,ThenthepossiblevaluesXcouldtake:

9e.g.2LetXbearandomv1010e.g.2

RandomvariableXrepresents“Themeasuringerrorsforsomemachineparts”.ThenthevaluesXcouldtakeis(a,b).e.g.1RandomvariableXrepresents“Thelengthoflifeforalamp”.(2)ContinuousIfallpossiblevaluesarandomvariablecouldtakewillfullyfillinanintervalontheaxis,thisvariablewillbecallacontinuousrandomvariable.ThenthevaluesXcouldtakeis10e.g.2RandomvariableXr1111Summary2.Twowaystoclassifyrandomvariable:discrete、continuous.1.Probabilitytheoryquantitativelyexaminestheinherentpatternofrandomphenomena,thusinordertoeffectivelysearchintorandomphenomena,wemustquantifyrandomevents.Whenrepresentingsomenon-numericalrandomeventwithnumbers，theconceptofrandomvariableisestablished.Therefore,randomvariableisdefinedasaspecialfunctioninthesamplespace.11Summary2.Twowaystoclas122.2Discreterandomvariables

Definition2.3Witheachpossibleoutcome,weassociateaNumbercalledtheprobabilityofxi·Thenumbersmustsatisfy(i)(ii)ArandomvariableXissaidtobeadiscreterandomvariableifitsrangespaceiseitherfiniteorcountablyinfinite,i.e.122.2Discreterandomvariables13Thenumbersmustsatisfy(i)(ii)Definition2.4

ThefunctionpiscalledtheprobabilitymassfunctioncalledtheprobabilitydistributionofX.(pmf)andthecollectionofpairs13Thenumbersmustsatisfy(i)(ii)14Example2.1Solution:LetXbearandomvariablewhosevaluesxarethepossiblenumbersofdefectivecomputers

Purchasedbytheschool.Thenxcanbeanyofthenumbers0,1,and2.Ashipmentof8similarmicrocomputerstoaretailoutletcontains3thataredefective.arandompurchaseof2ofthesecomputers,findtheprobabilitydistributionforthenumberofdefectives.Ifaschoolmakes14Example2.1Solution:LetXbea15Solution:LetXbearandomvariablewhosevaluesxarethepossiblenumbersofdefectivecomputers

Purchasedbytheschool.Thenxcanbeanyofthenumbers0,1,and2.ThustheprobabilitydistributionofXisx012p(x)15Solution:LetXbearandomvar162.3SomeImportantDiscrete

ProbabilityDistributionsUniformWehaveafinitesetofoutcomeswhichhasthesameprobabilityofoccurring(equallylikelyoutcomes).XissaidtohaveaUniformdistributionandweeachofWriteSo162.3SomeImportantDiscrete

17XissaidtohaveaUniformdistributionandweWritebulb,Example2.2Whenalightbulbisselectedatrandomfromaboxthatcontainsa40-wattbulb,a60-wattbulb,a75-wattanda100-wattbulb.FindSolution:eachelementofthesamplespaceoccurswithprobability1/4.Therefore,wehaveauniformdistribution:17XissaidtohaveaUniformdi18Solution:eachelementofthesamplespaceoccurswithprobability1/4.Therefore,wehaveauniformdistribution:Example2.3Whenadieistossed,S={1,2,3,4,5,6}.P(eachelementofthesamplespace)=1/6.Therefore,wehaveauniformdistribution,with18Solution:eachelementofthes19Example2.3Whenadieistossed,S={1,2,3,4,5,6}.P(eachelementofthesamplespace)=1/6.Therefore,wehaveauniformdistribution,withBernoullitrialABernoullitrialisanexperimentwhichhastwoLetp=P(success),q=P(failure)(q=1-p).‘success’and‘failure’.possibleoutcomes:19Example2.3Whenadieistosse20

ThepmfofXisBernoullitrialABernoullitrialisanexperimentwhichhastwoLetp=P(success),q=P(failure)(q=1-p).‘success’and‘failure’.possibleoutcomes:or20Bernoullimaterial

ThepmfofXisBernoullitr21Binomialeachofwhichmustresultineithera‘success’withConsiderasequenceofnindependentBernoullitrialsprobabilityofpora‘failure’withprobabilityq=1-p.LetX=thetotalnumberofsuccessesinthesentrialsothatXissaidtohaveaBinomialdistributionwithparametersP(thetotalnumberofxsuccesses

)=nandpandwewriteX~Bin(n,p)orX~b(x;n,p)Specialcase,21Binomialeachofwhichmustres22XissaidtohaveaBinomialdistributionwithparametersnandpandwewriteX~Bin(n,p)orX~b(x;n,p)Specialcase,whenn=1,wehaveWewriteB(n,p)b(1,p)22XissaidtohaveaBinomiald23BinomialDistribution23BinomialDistribution2324Theprobabilitythatacertainkindofcomponentwillsurviveagivenshocktestis3/4.Findtheprobabilitythatexactly2ofthenext4componentstestedsurvive.Example2.4AssumingthatthetestsareindependentandSolution:p=3/4foreachofthe4tests,weobtainExample2.5Theprobabilitythatapatientrecoversfromarareblooddiseaseis0.4.thisdisease,survive,If15peopleareknowntohavecontractedwhatistheprobabilitythat(a)atleast10(b)from3to8survive,and(c)exactly5survive?24Theprobabilitythatacertain25Theprobabilitythatapatientrecoversfromarareblooddiseaseis0.4.If15peopleareknowntohavecontractedthisdisease,whatistheprobabilitythat(a)atleast10survive,(b)from3to8survive,and(c)exactly5survive?Example2.5Solution：LetX=thenumberofpeoplethatsurvive.(a)(b)25Theprobabilitythatapatient26(c)(b)Example2.6(a)Theinspectoroftheretailerrandomlypicks20itemsAlargechainretailerpurchasesacertainkindofelectronicdevicefromamanufacturer.indicatesthatthedefectiverateofthedeviceis3%.Themanufacturer26(c)(b)Example2.6(a)Theinspe27fromashipment.(b)Supposethattheretailerreceives10shipmentsinaAlargechainretailerpurchasesacertainkindofelectronicdevicefromamanufacturer.Themanufacturerindicatesthatthedefectiverateofthedeviceis3%.Example2.6(a)Theinspectoroftheretailerrandomlypicks20itemsSolution：(a)DenotebyXthenumberofdefectiveDevicesamongthe20.ThenthisXfollowsab(x;20,0.03).willbeatleastonedefectiveitemamongthese20?Whatistheprobabilitythatthereshipmentscontainingatleastonedefectivedevice?shipment.monthandtheinspectorrandomlytests20devicesperWhatistheprobabilitythattherewillbe3Hence27fromashipment.(b)Supposet28Solution：(a)DenotebyXthenumberofdefectiveDevicesamongthe20.ThenthisXfollowsab(x;20,0.03).Hence(b)AssumingtheindependencefromshipmenttoTherefore,shipmentanddenotingbyY.Y=thenumberofshipmentscontainingatleastonedefective.ThenY~b(y;10,0.4562).28Solution：(a)DenotebyXthe29(b)AssumingtheindependencefromshipmenttoTherefore,shipmentanddenotingbyY.Y=thenumberofshipmentscontainingatleastonedefective.ThenY~b(y;10,0.4562).PoissonThepmfofarandomvariableXwhichhasaPoissondistributionwithparameterisgivenby29(b)Assumingtheindependence30PoissonThepmfofarandomvariableXwhichhasaPoissonandwewritedistributionwithparameterisgivenby30Poisson

materialPoissonThepmfofarandomvar3131NumberoftelephoneringsNumberoftrafficaccidentNumberofcustomersatreceptionEarthquakeVolcanicEruptionMassflooding31NumberoftelephoneringsNum32Poissondistribution32Poissondistribution3233单击图形播放/暂停ESC键退出Binomialdistribution

Poissondistribution33单击图形播放/暂停ESC键退出Binomialdistr34Duringalaboratoryexperimenttheaveragenumberofradioactiveparticlespassingthroughacounterin1millisecondis4.Whatistheprobabilitythat6articlesenterthecounterinagivenmillisecond?Example2.7Solution：UsingthePoissondistributionwithx=6and,WefindfromTable1thatExample2.8Tenistheaveragenumberofoiltankersarrivingeachdayacertainportcity.Thefacilitiesattheportcan34Duringalaboratoryexperiment35handleatmost15tankersperday.Example2.8Tenistheaveragenumberofoiltankersarrivingeachdayacertainportcity.ThefacilitiesattheportcanSolution：LetXbethenumberoftankersarrivingeachday.Then,Whatistheprobabilitythatonagivendaytankershavetobeturnedaway?usingTable,wehave35handleatmost15tankersper362.4CumulativeDistributionFunctionsDefinition2.5Thecumulativedistributionfunction(cdf)ofthe

randomvariableXisdefinedtobeandisdenotedbyF(x).PropertiesofF(x):(i)Fisnon-decreasing.(ii)i.e.i.e.if362.4CumulativeDistributionFu37PropertiesofF(x):(i)Fisnon-decreasing.i.e.if(ii)(iii)Fisrightcontinuous.(iv)F(x)isdefinedforallrealnumbersx.

ThecdfofadiscreterandomvariableXisastepfunctionwithjumpsatthei.e.37PropertiesofF(x):(i)Fis38ThecdfofadiscreterandomvariableXisastepfunctionwithjumpsatthe38Thecdfofadiscreterandomv39e.g.39e.g.3940e.g.Example2.9ThepmfofXis40e.g.Example2.9ThepmfofXis41ThepmfofXisFind:1)TheCumulativedistributionfunctionofX.2)X-123Solution:1)-123Example2.941ThepmfofXisFind:1)TheCum42Solution:1)-1232)42Solution:1)-1432)-101231432)-101442.5ContinuousRandomVariablesDefinition2.6XisacontinuousrandomvariableifthereexistsaThefunctionfiscalledtheprobabilitydensityfunctionwiththepropertythatforeverysubsetofrealnonnegativefunctionfdefinedforallrealxnumbersB(pdf)ofX.Propertiesofthepdf(i)442.5ContinuousRandomVariable45Propertiesofthepdf(i)ThisfollowsbysettingB=(ii)145ThisfollowsfromPropertiesofthepdf(i)Thisf46(iii)IfweletB=[a,b]thenThisfollowsfrom(iv)Thisfollowsfrom461(iii)IfweletB=[a,b]t47(iv)ThisfollowsfromNotes(a)IfXiscontinuousthenF(x)iscontinuous.Also,(b)P(a≤X≤b)representsthebetweenx=aandx=b.areaunderthegraphoff

f(x)=F’(x)

atallpointwhereFiscontinuous.47(iv)ThisfollowsfromNotes(a)48f(x)=F’(x)(c)Themeaningofdensityfunction:i.e.

TheprobabilitythatXisinasmallintervalisapproximatelyequaltof(x)timesthewidthoftheinterval(d)ForanyspecifiedvalueofX,sayx0,wehave.(b)P(a≤X≤b)representsthebetweenx=aandx=b.areaunderthegraphoff

48f(x)=F’(x)(c)Themeaningofd49i.e.

TheprobabilitythatXisinasmallintervalisapproximatelyequaltof(x)timesthewidthoftheinterval(d)ForanyspecifiedvalueofX,sayx0,wehave.Hence,thenifXiscontinuousthentheprobabilities49i.e.TheprobabilitythatXis50

Example2.10LetXbeacontinuousr.v.with.pdfFindSolution:50Example2.10FindSolution:50512.6SomeContinuousProbabilityDistributionsUniform(orrectangular)DistributionAuniformrandomvariableXontheinterval(a,b)hasprobabilitydensityfunction(pdf)WewriteX~U(a,b).512.6SomeContinuousProbabilit52Cdf:52Cdf:5253Example2.11

Supposethatalargeconferenceroomforacertaincompanycanbereservedfornomorethan4hours,However,theuseoftheconferenceroomissuchthatNormalDistributionThepdfofaNormalrandomvariable,X,withisgivenbyparametersbothlongandshortconferencesoccurquiteoften.Infact,itcanbeassumedthatlengthXofaconferencehasauniformdistributionontheinterval[0,4].(a)Whatistheprobabilitydensityfunction?(b)Whatistheprobabilitythatanygivenconferencelastsatleast3hours?53Example2.11Supposethatala54NormalDistributionThepdfofaNormalrandomvariable,X,withisgivenbyparametersWewrite54NormalDistributionThepdfof55GeometricCharacteristicsofthedensityforNormalDistribution55GeometricCharacteristicsoft56565657575758Cdf:58Cdf:5859NormalDistributionisoneofthemostimportantandCommonlyobserveddistribution.Forexample,measuringuncertainty,humanphysicalcharacteristicssuchasheight,weight,etc…MeasurementsformanufacturedproductsmadeundersameConditions,i.e.length,diameter,mass,height,etc…,allseemtoobeynormaldistribution.ApplicationandBackgroundoftheNormalDistribution

Gaussianmaterial59NormalDistributionis60InthespecialcasewhenandthedistributioniscalledthestandardNormaldistribution.i.e.Wewrite60Inthespecialcasewhen61ForastandardNormalrandomvariableXdistributioniscalledthestandardNormaldistribution.i.e.Wewrite61ForastandardNormalrandomv62Example2.12Givenastandardnormaldistribution，findtheareaunderthecurvethatlies(a)totherightofz=1.84(b)betweenz=-1.97andz=0.86.Example2.13Givenastandardnormaldistribution，findthevalueofksuchthat62Example2.12Givenastandardn63Figure2.10AreasforExample2.16(a)P(Z>k)=0.3015，Example2.13Givenastandardnormaldistribution，findthevalueofksuchthatATheorem:IfThenand(b)P(k<Z<-0.18)=0.4197.63Figure2.10AreasforExample264ForaNormalrandomvariableXATheorem:IfThenExample2.1464ForaNormalrandomvariableX65Example2.14GivenarandomvariableXhavinganormaldistributionwith，FindtheprobabilitythatXassumesavaluebetween45and62.Example2.15GiventhatXhasanormaldistributionwithand，findtheprobabilitythatXassumesavaluegreaterthan362.ExponentialDistributionThepdfofanExponentialrandomvariablewithParameterisgivenby65Example2.14Givenarandomvar66ExponentialDistributionThepdfofanExponentialrandomvariablewithParameterisgivenbyWewrite66ExponentialDistributionThepd67Somecomponentsordeviceshavealifespanthatobeystheexponentialdistribution.Forexample,lifespanofmobiledevices,electricpowerdevices,animals,andothersallobeythisexponentialdistribution.ApplicationandBackgroundCdf:67Somecomponentsordev68If5ofthesecomponentsareinstalledindifferentSupposethatasystemcontainsacertaintypeofcomponentwhosetimeinyearstofailureisgivenbyT.TherandomvariableTismodelednicelybytheExample2.16exponentialdistributionwithmeantimetofailureSystemswhatistheprobabilitythatatleast2arestillfunctioningattheendof8years?68If5ofthesecomponentsarei692.7FunctionsofRandomVariablesa)whenther.v.xisdiscreteThepmfofYtakesonagivenvalue,sayyj,isSupposether.v.xhastheprobabilitydistributionbelowExample2.17xi012345P(xi)p(0)

p(1)p(2)p(3)p(4)p(5)LetY=(X-2)2,Findthepmfofther.v.Y.692.7FunctionsofRandomVariab70Supposether.v.xhastheprobabilitydistributionbelowExample2.17xi012345P(xi)p(0)

p(1)p(2)p(3)p(4)p(5)LetY=(X-2)2,Findthepmfofther.v.Y.Solution:01