章伯分怖章伯分彳布（WeibulldistribulM指譴攵分彳布舄一特例。其p.d.泰）f（x）=o徹尸一七一心"注＞Oj其中a,B＞0。以陽3向表此分布，有二參敷a,們潟尺度＞?,。舄形狀參敷。若取。二1,劇」得的）分布，以職電1）=￡四表之。底下富合出一些菖白分布p.d.之圈形。章伯分布是瑞典物理孥家WaloddiWeibullSS展弓鼠化材料的理^,於西元1939年所引逵，是一重交新的分布。在可靠度理^及有^嘉命梭定冏題裨，常少不了章伯分布的影子。網(wǎng)（土月）分布的分布函敷SF（對=1—廠心丘注＞0令期望值典燮巽敷分別Se（x）=q-/t（i+?國X）=a-2^（r（i+1）-（r（i十i））2）

CharacteristicEffectsoftheShapeParameter,p,fortheWeibullDistribution1.-互TheWeibullshapeparameter,P,isalsoknownastheslope.ThisisbecausethevalueofPisequaltotheslopeoftheregressedlineinaprobabilityplot.Differentvaluesoftheshapeparametercanhavemarkedeffectsonthebehaviorofthedistribution.Infact,somevaluesoftheshapeparameterwillcausethedistributionequationstoreducetothoseofotherdistributions.Forexample,whenp=1,thepdfofthethree-parameterWeibullreducestothatofthetwo-parameterexponentialdistributionor:where慧R"w"failurerate.1.-互Theparameterpisapurenumber,i.e.itisdimensionless.TheEffectofponthepdfFigure6-1showstheeffectofdifferentvaluesoftheshapeparameter,p,ontheshapeofthepdf.Onecanseethattheshapeofthepdfcantakeonavarietyofformsbasedonthevalueofp.

DLHULIDLILISUDLID4UD1102UV'/eibullD^fV'/ithLKp<1. .andp>1Tirne(t)LILHUUDDDSLIDDD4UDDD2U\'VeibullDiJ'fwithU<p<1,[\=],andp>1Figure6-1:TheeffectoftheWeibullshapeparameteronthepdf.For0<DLHULIDLILISUDLID4UD1102UV'/eibullD^fV'/ithLKp<1. .andp>1Tirne(t)LILHUUDDDSLIDDD4UDDD2U\'VeibullDiJ'fwithU<p<1,[\=],andp>1Figure6-1:TheeffectoftheWeibullshapeparameteronthepdf.For0<As繇丁一皈一。皿》),A’w-收手，顧何-OCT-Q.f(?decreasesmonotonicallyandisconvexasTincreasesbeyondthevalueof1.Themodeisnon-existent.ForP>1:

.f(T=0atT=0(orY).?f(Tincreasesas酸''(themode)anddecreasesthereafter.. For&<2.6theWeibullpdfispositivelyskewed(hasarighttail),for2.6<&<3.7itscoefficientofskewnessapproacheszero(notail).Consequently,itmayapproximatethenormalpdf,andfor&>3.7itisnegativelyskewed(lefttail).Thewaythevalueof&relatestothephysicalbehavioroftheitemsbeingmodeledbecomesmoreapparentwhenweobservehowitsdifferentvaluesaffectthereliabilityandfailureratefunctions.Notethatfor&=0.999,f(0)=酸030:3,butfor&=1.001,f(°)=0.ThisabruptshiftiswhatcomplicatesMLEestimationwhen&isclosetoone.TheEffectofflonthecdfandReliabilityFunctionEffectuf-A^lbullShspHParam玳日「口nProbsNIlf：Effectuf-A^lbullShspHParam玳日「口nProbsNIlf：PlotFigure6-2:EffectofflonthecdfonaWeibullprobabilityplotwithafixedvalueof〃.Figure6-2showstheeffectofthevalueof&onthecdf,asmanifestedintheWeibullprobabilityplot.Itiseasytoseewhythisparameterissometimesreferredtoastheslope.Notethatthemodelsrepresentedbythethreelinesallhavethesamevalueof〃.Figure6-3showstheeffectsofthesevariedvaluesof&onthereliabilityplot,whichisalinearanalogoftheprobabilityplot.

U1LLI.UU2UULILI3DDA0400DO5DDA06UUDO7DDUDTirrt!.(t)Figure6-3:TheeffectofvaluesofpontheWeibullreliabilityplot.U1LLI.UU2UULILI3DDA0400DO5DDA06UUDO7DDUDTirrt!.(t)Figure6-3:TheeffectofvaluesofpontheWeibullreliabilityplot.R(Ddecreasessharplyandmonotonicallyfor0<&<1andisconvex.For&=1,R(T)decreasesmonotonicallybutlesssharplythanfor0<&<1andisconvex.For&>1,R(T)decreasesasTincreases.Aswear-outsetsin,thecurvegoesthroughaninflectionpointanddecreasessharply.TheEffectofpontheWeibullFailureRateFunctionThevalueof&hasamarkedeffectonthefailurerateoftheWeibulldistributionandinferencescanbedrawnaboutapopulation'sfailurecharacteristicsjustbyconsideringwhetherthevalueof&islessthan,equalto,orgreaterthanone.

ULI2ULIULI18UULiltiUU1H4LIUL1120ULI1LIUULIUSUULIU6ULILIU40ULIU2LIUTirre,it)Figure6-4:TheeffectofBontheWeibullfailureratefunction.ULI2ULIULI18UULiltiUU1H4LIUL1120ULI1LIUULIUSUULIU6ULILIU40ULIU2LIUTirre,it)Figure6-4:TheeffectofBontheWeibullfailureratefunction.AsindicatedbyFigure6-4,populationswith&<1exhibitafailureratethatdecreaseswithtime,populationswithP=1haveaconstantfailurerate(consistentwiththeexponentialdistribution)andpopulationswith&>1haveafailureratethatincreaseswithtime.AllthreelifestagesofthebathtubcurvecanbemodeledwiththeWeibulldistributionandvaryingvaluesof&.TheWeibullfailureratefor0<&<1isunboundedatT=0(or?).Thefailurerate,久⑺,decreasesthereaftermonotonicallyandisconvex,approachingthevalueofzeroas繇'一一°°or4(霞^303)=0.Thisbehaviormakesitsuitableforrepresentingthefailurerateofunitsexhibitingearly-typefailures,forwhichthefailureratedecreaseswithage.Whenencounteringsuchbehaviorinamanufacturedproduct,itmaybeindicativeofproblemsintheproductionprocess,inadequateburn-in,substandardpartsandcomponents,orproblemswithpackagingandshipping.

ForP=1,^(^)yieldsaconstantvalueof繇豆"or:警=a=ia(q=a=*Thismakesitsuitableforrepresentingthefailurerateofchance-typefailuresandtheuseful肝eperiodfailurerateofunits.Forp>1,^(T)increasesasTincreasesandbecomessuitableforrepresentingthefailurerateofunitsexhibitingwear-outtypefailures.For1<p<2,the4(T)curveisconcave,consequentlythefailurerateincreasesatadecreasingrateasTincreases.Forp=2thereemergesastraightlinerelationshipbetween4(T)andT,startingatavalueof從T)=0atT=y,andincreasingthereafterwithaslopeofConsequently,thefailurerateincreasesataconstantrateasincreasingthereafterwithaslopeofConsequently,thefailurerateincreasesataconstantrateasincreases.Furthermore,if"=1theslopebecomesequalto2,andwheny=0,4(T)becomesastraightlinewhichpassesthroughtheoriginwithaslopeof2.Notethatatp=2,theWeibulldistributionequationsreducetothatoftheRayleighdistribution.Whenp>2,the^(T)curveisconvex,withitsslopeincreasingasTincreases.Consequently,thefailurerateincreasesatanincreasingrateasTincreasesindicatingwear-outlife.TopCharacteristicEffectsoftheScaleParameter,n,fortheWeibullDistributionFigure6-5:TheeffectsofnFigure6-5:TheeffectsofnontheWeibullpdfforacommonp.Achangeinthescaleparameter"hasthesameeffectonthedistributionasachangeoftheabscissascale.Increasingthevalueof"whileholdingPconstanthastheeffectofstretchingoutthepdf.Sincetheareaunderapdfcurveisaconstantvalueofone,the"peak"ofthepdfcurvewillalsodecreasewiththeincreaseof〃，asindicatedinFigure6-5.Ifnisincreasedwhilepandyarekeptthesame,thedistributiongetsstretchedouttotherightanditsheightdecreases,whilemaintainingitsshapeandlocation.Ifnisdecreasedwhilepandyarekeptthesame,thedistributiongetspushedintowardstheleft(i.e.towardsitsbeginningortowards0or?),anditsheightincreases.nhasthesameunitsasT,suchashours,miles,cycles,actuations,etc.TopCharacteristicEffectsoftheLocationParameter,yfortheWeibullDistributionThelocationparameter,Yasthenameimplies,locatesthedistributionalongtheabscissa.Changingthevalueofyhastheeffectof"sliding"thedistributionanditsassociatedfunctioneithertotheright(ify>0)ortotheleft(ify<0).EffectofLocationparameteryonWeibull網(wǎng)EffectofLocationparameter7onWeibullEffectofLocationparameteryonWeibull網(wǎng)EffectofLocationpar