概论理论的基础.ppt

elementsofprobabilitytheory dr daganglu professorschoolofcivilengineeringharbininstituteoftechnology 1randomvariables 1 1 1definitionofrandomvariables letbearandomexperiment isitssamplespace for thereexistsarealsingle valuefunction iffor isarandomeventoneventfield thenwecancallarandomvariable or 1 2descriptionofrandomvariables 1 probabilitymassfunction pmf fordiscreterandomvariable therearethreekindsoffunctionsfordescriptionofrandomvariablesasfollows 2 cumulativedistributionfunction cdf fordiscreterandomvariableforcontinuousrandomvariable 3 probabilitydensityfunction pdf forcontinuousrandomvariable 1randomvariables 2 1randomvariables 3 1 probabilitymassfunction pmf theprobabilitymassfunction pmf isdefinedfordiscreterandomvariablesasfollows representsprobabilitythatadiscreterandomvariableisequaltoaspecificvalue whereisarealnumber mathematically 2 cumulativedistributionfunction cdf representsthetotalsum orintegral ofallprobabilityfunctions continuousanddiscrete correspondingtovalueslessthanorequalto mathematically thecumulativedistributionfunction cdf isdefinedforbothdiscreteandcontinuousrandomvariablesasfollows 1randomvariables 4 3 probabilitydensityfunction pdf forcontinuousrandomvariables theprobabilitydensityfunction pdf isdefinedasthefirstderivativeofthecumulativefunction mathematically propertiesofcdf pdfandpmf thecdfisapositive nondecreasingfunctionwhosevalueisbetween0and1 if then forcontinuousrandomvariable 1randomvariables 5 1 3momentsofrandomvariables 1 meanorexpectedvalue firstmoment themeanvalueofarandomvariableisdenotedby foracontinuousrandomvariable themeanvalueisdefinedas foradiscreterandomvariable themeanvalueisdefinedas theexpectedvalueofiscommonlydenotedbyandisequaltothemeanvalueof 1randomvariables 6 foradiscreterandomvariable thenthmomentisdefinedas foracontinuousrandomvariable thenthmomentisdefinedas theexpectedvalueofiscalledthenthmomentof themathematicalexpectationofanarbitraryfunctionoftherandomvariableisdefinedas 1randomvariables 7 thestandarddeviationofisdefinedasthepositivesquarerootofthevariance animportantformula thevarianceofarandomvariableisameasureofthedegreeofrandomnessaboutthemeanvalue 2 varianceandstandarddeviation secondmoment 1randomvariables 8 ifasetofnobservationsareobtainedforaparticularrandomvariable then thenon dimensionalcoefficientofdeviationisdefinedasthestandarddeviationdividedbythemean 3 momentsofsample thetruemeancanbeapproximatedbythesamplemean thetruestandarddeviationcanbeapproximatedbythesamplestandarddeviation 1randomvariables 9 1 4standardformofrandomvariables letbearandomvariable thestandardformof denotedby isdefinedas themeanvalueofiscalculatedasfollows thedeviationofiscalculatedasfollows 2commonprobabilitymodels 1 2 1uniformrandomvariables 2commonprobabilitymodels 2 2 2normalrandomvariables forastandardnormalrvu forageneralnormalrvx thepdfofuisdenotedby thecdfofuisdenotedby 2commonprobabilitymodels 3 standardnormalrandomvariables relationshipbetweengeneralnormalrvxandstandardnormalrvu 2commonprobabilitymodels 4 propertiesofdistributionfunctionofanormalrv thepdfissymmetricalaboutthemeanvalue thesumofandisequalto1 theinversecdfofanormalrv 2commonprobabilitymodels 5 2 3lognormalrandomvariables cdf pdflognormalrvs definitionoflognormalrvs 2commonprobabilitymodels 6 momentsoflognormalrvs if then propertiesoflognormalrvs 2commonprobabilitymodels 7 2 4gammadistribution pdfofgammarvs gammafunction for momentsofgammarvs aredistributionparameters 2commonprobabilitymodels 8 2 5extremetype gumbeldistribution cdf pdfofextreme rvs for momentsofextreme rvs aredistributionparameters 2commonprobabilitymodels 9 2 6extremetype cdf pdfofextreme rvs for momentsofextreme rvs aredistributionparameters 2commonprobabilitymodels 10 2 7extremetype weibulldistribution cdfofthelargestvalues for momentsofthelargestvalues aredistributionparameters 2commonprobabilitymodels 11 2 7extremetype weibulldistribution cdfofthesmallestvalues for momentsofthesmallestvalues aredistributionparameters 2commonprobabilitymodels 12 2 8poissondistribution propertiesofpoissondistribution assumptionsofpoissondistribution itisadiscreteprobabilitydistributionitcanbeusedtocalculatethepmfforthenumberofoccurrenceofaparticulareventinatimeorspaceinterval 0 t theoccurrenceofeventsareindependentofeachothertwoormoreeventscannotoccursimultaneously pmfofpoissondistribution representsthemeanoccurrencerateoftheeventwhichisusuallyobtainedfromstatisticaldata representsthenumberofoccurrencesofaneventwithinaprescribedtime orspace interval 0 t 2commonprobabilitymodels 13 2 8poissondistribution momentsofpoissondistribution thereturnperiodofpoissondistribution theannualoccurrenceprobabilityofpoissondistribution cdfofpoissondistribution 3randomvectors 1 3 1definitionofrandomvectors arandomvectorisdefinedasavector orset ofrandomvariables 3 2thejointcdfandpdfofrandomvectors thejointcumulativedistributionfunction thejointprobabilitydistributionfunction forcontinuousrvs fordiscretervs 3randomvectors 2 3 3marginaldensityfunctionofrandomvectors forcontinuousrandomvariables amarginaldensityfunction mdf foreachisdefinedas 3 4casesofjointcdfandpdfoftwocontinuousrvs thejointcdfofxandy thejointpdfofxandy themdfofxandy 3randomvectors 3 3 5conditionaldistributionfunctionofrandomvectors forcontinuousrandomvariables theconditionaldistributionfunctionforarandomvector x y isdefinedas 3 6statisticalindependenceofrandomvectors iftherandomvariablesxandyarestatisticalindependent then 3randomvectors 4 3 7correlationofrandomvariables 1 covarianceoftworvs 2 coefficientofcorrelation theformulaofcorrelationcoefficient 1 2 fortwocontinuousvariablesxandy 3randomvectors 5 propertiesofcorrelationcoefficient 1 2 thevaluesofindicatesthedegreeoflineardependencebetweenthetworandomvariablesxandy ifiscloseto1 thenxandyarelinearlyrelatedtoeachother ifiscloseto0 thenxandyarenotlinearlyrelatedtoeachother differencebetweenuncorrelatedandstatisticalindependent xandyareuncorrelated xandyarestatisticalindependent statisticalindependentisamuchstrongerstatementthanuncorrelated 3randomvectors 6 3 covariancematrixofrandomvectors forarandomvectorwithnrandomvariables thecovariancematrixisdefinedas thematrixofcorrelationefficientisdefinedas 3randomvectors 7 propertiesofand 1 symmetricmatrices 2 thediagonalterms 3 ifallnrandomvariablesareuncorrelated then 3randomvectors 8 statisticalestimateofthecorrelationcoefficient assumethattherearenobservationsofvariablexandnobservationsofvariabley samplemean samplestandarddeviation sampleestimateofthecorrelationcoefficient 4functionsofrandomvariables 1 1 linearfunctionsofrandomvariables where theareconstants letybealinearfunctionofrandomvariables momentsoflinearfunctionsofrandomvariables 4functionsofrandomvariables 2 varianceoflinearfunctionsofuncorrelatedrandomvariables ifthenrandomvariablesareuncorrelatedwitheachother then for propertiesoflinearfunctionsofrandomvariables 1 theprobabilitydistributionsoftherandomvariablesarenotneeded 2 thelinearfunctionyofuncorrelatednormalrandomvariablesisanormalrandomvariablewithdistributionparametersand 3 theconstantdoesnotaffectthevariance butitdoesaffectthemeanvalue 4functionsofrandomvariables 3 2 productoflognormalrandomvariables letybeafunctioninvolvingtheproductsofseveralrandomvariables assumethattheserandomvariablesarestatisticalindependent lognormalrandomvariables theaboveformularepresentsthesumofnormallydistributedrandomvariables thequantityisanormallydistributedrandomvariable isalognormallydistributedrandomvariable 4functionsofrandomvariables 5 momentsofthelognormallydistributedrandomvariabley 4functionsofrandomvariables 6 3 nonlinearfunctionsofrandomvariables letybeageneralnonlinearfunctionoftherandomvariables thefirstordertaylorseriesexpansionofy momentsofnonlinearfunctiony mathematically where iscalled designpoint whichisdenotedby forareuncorrelated 5centrallimittheorems 1 letthefunctionybethesumofnstatisticallyindependentrandomvariableswhoseprobabilitydistributionsarearbitrary thecentrallimittheorystatesthatasnapproachesinfinity thesumoftheseindependentrandomvariablesapproachesanormalprobabilitydistributionifnoneoftherandomvariablestendstodominatethesum assumptions theorem ifwehaveafunctiondefinedasthesumofalargenumberofrandomvariables thenwewouldexpectthesumtobeapproximatelyasanormallydistributed conclusions thesumofvariablesisoftenusedtomodelthetotalloadonastructure therefore thetotalloadcanbeapproximatedasanormalvar