韋伯分佈韋伯分佈(Weibulldistribution)以指數分佈為一特例。其p.d.f.為其中α,β>0。以表此分佈,有二參數α,β,α為尺度參數,β為形狀參數。若取β=1,則得分佈,以表之。底下給出一些韋伯分佈p.d.f.之圖形。韋伯分佈是瑞典物理學家WaloddiWeibull,為發展強化材料的理論,於西元1939年所引進,是一較新的分佈。在可靠度理論及有關壽命檢定問題裡,常少不了韋伯分佈的影子。分佈的分佈函數為期望值與變異數分別為CharacteristicEffectsoftheShapeParameter,β,fortheWeibullDistributionTheWeibullshapeparameter,β,isalsoknownastheslope.Thisisbecausethevalueofβisequaltotheslopeoftheregressedlineinaprobabilityplot.Differentvaluesoftheshapeparametercanhavemarkedeffectsonthebehaviorofthedistribution.Infact,somevaluesoftheshapeparameterwillcausethedistributionequationstoreducetothoseofotherdistributions.Forexample,whenβ=1,thepdfofthethree-parameterWeibullreducestothatofthetwo-parameterexponentialdistributionor:wherefailurerate.Theparameterβisapurenumber,i.e.itisdimensionless.TheEffectofβonthepdfFigure6-1showstheeffectofdifferentvaluesoftheshapeparameter,β,ontheshapeofthepdf.Onecanseethattheshapeofthepdfcantakeonavarietyofformsbasedonthevalueofβ.Figure6-1:TheeffectoftheWeibullshapeparameteronthepdf.For0<β1:As(orγ),As,.f(T)decreasesmonotonicallyandisconvexasTincreasesbeyondthevalueofγ.Themodeisnon-existent.Forβ>1:f(T)=0atT=0(orγ).f(T)increasesas(themode)anddecreasesthereafter.Forβ<2.6theWeibullpdfispositivelyskewed(hasarighttail),for2.6<β<3.7itscoefficientofskewnessapproacheszero(notail).Consequently,itmayapproximatethenormalpdf,andforβ>3.7itisnegativelyskewed(lefttail).Thewaythevalueofβrelatestothephysicalbehavioroftheitemsbeingmodeledbecomesmoreapparentwhenweobservehowitsdifferentvaluesaffectthereliabilityandfailureratefunctions.Notethatforβ=0.999,f(0)=,butforβ=1.001,f(0)=0.ThisabruptshiftiswhatcomplicatesMLEestimationwhenβisclosetoone.TheEffectofβonthecdfandReliabilityFunctionFigure6-2:EffectofβonthecdfonaWeibullprobabilityplotwithafixedvalueofη.Figure6-2showstheeffectofthevalueofβonthecdf,asmanifestedintheWeibullprobabilityplot.Itiseasytoseewhythisparameterissometimesreferredtoastheslope.Notethatthemodelsrepresentedbythethreelinesallhavethesamevalueofη.Figure6-3showstheeffectsofthesevariedvaluesofβonthereliabilityplot,whichisalinearanalogoftheprobabilityplot.Figure6-3:TheeffectofvaluesofβontheWeibullreliabilityplot.R(T)decreasessharplyandmonotonicallyfor0<β<1andisconvex.Forβ=1,R(T)decreasesmonotonicallybutlesssharplythanfor0<β<1andisconvex.Forβ>1,R(T)decreasesasTincreases.Aswear-outsetsin,thecurvegoesthroughaninflectionpointanddecreasessharply.TheEffectofβontheWeibullFailureRateFunctionThevalueofβhasamarkedeffectonthefailurerateoftheWeibulldistributionandinferencescanbedrawnaboutapopulation'sfailurecharacteristicsjustbyconsideringwhetherthevalueofβislessthan,equalto,orgreaterthanone.Figure6-4:TheeffectofβontheWeibullfailureratefunction.AsindicatedbyFigure6-4,populationswithβ<1exhibitafailureratethatdecreaseswithtime,populationswithβ=1haveaconstantfailurerate(consistentwiththeexponentialdistribution)andpopulationswithβ>1haveafailureratethatincreaseswithtime.AllthreelifestagesofthebathtubcurvecanbemodeledwiththeWeibulldistributionandvaryingvaluesofβ.TheWeibullfailureratefor0<β<1isunboundedatT=0(orγ).Thefailurerate,λ(T),decreasesthereaftermonotonicallyandisconv...