韦伯分布.docx

韦伯分布.docx

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韦伯分布

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,fortheWeibullDistribution

TheWeibullshapeparameter,β,isalsoknownastheslope.Thisisbecausethevalueofβisequaltotheslopeoftheregressedlineinaprobabilityplot.Differentvaluesoftheshapeparametercanhavemarkedeffectsonthebehaviorofthedistribution.Infact,somevaluesoftheshapeparameterwillcausethedistributionequationstoreducetothoseofotherdistributions.Forexample,whenβ=1,thepdfofthethree-parameterWeibullreducestothatofthetwo-parameterexponentialdistributionor:

wherefailurerate.

Theparameterβisapurenumber,i.e.itisdimensionless.

TheEffectofβonthepdf

Figure6-1showstheeffectofdifferentvaluesoftheshapeparameter,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.

TheEffectofβonthecdfandReliabilityFunction

Figure6-2:

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.

TheEffectofβontheWeibullFailureRateFunction

ThevalueofβhasamarkedeffectonthefailurerateoftheWeibulldistributionandinferencescanbedrawnaboutapopulation'sfailurecharacteristicsjustbyconsideringwhetherthevalueofβislessthan,equalto,orgreaterthanone.

Figure6-4:

TheeffectofβontheWeibullfailureratefunction.

AsindicatedbyFigure6-4,populationswithβ<1exhibitafailureratethatdecreaseswithtime,populationswithβ=1haveaconstantfailurerate（consistentwiththeexponentialdistribution）andpopulationswithβ>1haveafailureratethatincreaseswithtime.AllthreelifestagesofthebathtubcurvecanbemodeledwiththeWeibulldistributionandvaryingvaluesofβ.

TheWeibullfailureratefor0<β<1isunboundedatT=0（orγ）.Thefailurerate,λ（T）,decreasesthereaftermonotonicallyandisconvex,approachingthevalueofzeroasorλ（）=0.Thisbehaviormakesitsuitableforrepresentingthefailurerateofunitsexhibitingearly-typefailures,forwhichthefailureratedecreaseswithage.Whenencounteringsuchbehaviorinamanufacturedproduct,itmaybeindicativeofproblemsintheproductionprocess,inadequateburn-in,substandardpartsandcomponents,orproblemswithpackagingandshipping.

Forβ=1,λ（T）yieldsaconstantvalueofor:

Thismakesitsuitableforrepresentingthefailurerateofchance-typefailuresandtheusefullifeperiodfailurerateofunits.

Forβ>1,λ（T）increasesasTincreasesandbecomessuitableforrepresentingthefailurerateofunitsexhibitingwear-outtypefailures.For1<β<2,theλ（T）curveisconcave,consequentlythefailurerateincreasesatadecreasingrateasTincreases.

Forβ=2thereemergesastraightlinerelationshipbetweenλ（T）andT,startingatavalueofλ（T）=0atT=γ,andincreasingthereafterwithaslopeof.Consequently,thefailurerateincreasesataconstantrateasTincreases.Furthermore,ifη=1theslo