ch05-discrete and continuous probability distributions

CHAPTER5 DiscreteandContinuousProbabilityDistributions toaccompanyBusinessStatistics ADecision MakingApproach6thedition byDavidF Groebner Chapter5 LearningObjectives Distinguishbetweendiscreteandcontinuousrandomvariables DifferentiatebetweenthebinomialandthePoissondiscreteprobabilitydistributionsandtheirapplications Beabletodiscusstheimportantpropertiesofthenormalprobabilitydistribution Recgonizesituationsinwhichtheuniformandexponentialdistributionsapply 5 1TheBinomialProbabilityDistribution5 2OtherDiscreteProbabilityDistributions5 3TheNormalProbabilityDistribution5 4OtherContinuousProbabilityDistributions 5 1TheBinomialProbabilityDistribution 5 1 1Vocabularies词汇5 1 2MainContent主要内容5 1 3Terminologies专业词汇5 1 4Afterschoolextend课外扩展 5 1 1Vocabularies 1 strain 1 Thecollectivedescendantsofacommonancestor arace stock line orbreed 家系 血统 一个有共同祖先的集体后裔 家系 家谱 世系或血统 2 Anyofthevariouslinesofancestryunitedinanindividualorafamily ancestryorlineage 品系 联合在个人或家族上的任何一个祖先的血系 血统或家系 3 Akindorsort 一种或一类 5 1 1Vocabularies 2 DouglasAmunicipalboroughofsoutheastIsleofMan England ontheIrishSea Itisapopularseasideresortandthemostimportantcityontheisland Population 20 368 道格拉斯 爱尔兰海上英国马恩岛东南部自治市 它是岛上最重要的城市和有名的海滨胜地 人口20 368 男子名 5 1 1Vocabularies 3 cloneAreplicaofaDNAsequence suchasagene producedbygeneticengineering DNA复制品 克隆4 tread 1 Togoonfoot walk 步行 走 2 Tosetdownthefoot step 踏 放下脚步 踏步 3 Thepartofawheelortirethatmakescontactwiththeroadorrails 踏面 车轮或轮胎与路面或轨道接触的部分 4 Thepartofashoesolethattouchestheground 鞋底 鞋底接触地面的部分 5 1 1Vocabularies 5 vehicle 1 Adeviceorstructurefortransportingpersonsorthings aconveyance 交通工具 运输人或物的器具或结构 交通工具 Eg aspacevehicle 航天器 2 Aself propelledconveyancethatrunsontires amotorvehicle 机动车辆 汽车 一种装有轮胎的自行驱动的交通工具 机动车辆 3 Amediumthroughwhichsomethingistransmitted expressed oraccomplished 传播媒介 用来传递 表达或完成某物的工具或手段 Eg Hisnovelsareavehicleforhispoliticalviews 他的小说是传达他的政治观点的媒介 5 1 1Vocabularies 6 wear磨损常与away down或off一起使用 Eg rockswornawaybythesea shoesworndownattheheels 被海水冲刷的岩石 跟部磨损的鞋子7 involve含有 包括 使参与 影响 beinvolvedin包含在 与 有关 被卷入 专心地 做 beinvolvedwith涉及 becomeinvolvedin卷入 陷入 getinvolvedwith给 缠住Eg Thematterisseriousbecauseitinvolvesyourreputation Hewasinvolvedinworkingoutaplan 5 1 1Vocabularies 8 quasi Tosomedegree insomemanner 表示 类似 在某种程度上 在某种意义上9 item 1 Asinglearticleorunitinacollection anenumeration oraseries 项目 细目 文集 细目或一套中的单独条目或项目 2 Aclauseofadocument suchasabillorcharter 条款 文件中的条款 例如一个议案或者是一个章节 3 Anentryinanaccount 帐目中的一个条目 4 Abitofinformation adetail 一些信息 细节 5 Ashortpieceinanewspaperormagazine 短文 报纸或杂志的一小段 5 1 1Vocabularies 10 CatalogSale目录销售 消费者通过查阅 目录购物商场 定期发行的购物目录 拨打 商场 话务中心的电话订购 再由专业快递公司提供快捷优质的送货上门服务 消费者经过验货甚至试用之后付款的购物方式 目录销售于50年前在经济发达国家中率先兴起 如今已成为国际社会比较流行的新型商业形式 根据美国1980和1990年代两次经济萧条时期的经验 目录销售和仓储超市是两种能够在经济不景气中唯一取得成功的零售业形式 随着国际互联网的出现 目录销售已经从邮寄印刷品发布商品信息发展成为利用因特网进行在线商品信息传播的方式 5 1 1Vocabularies personnelmanager 人事主管 dismantle 分解 sideeffects 副作用 bloodpressure 血压 clinical 临床的 5 1 2MainContent 1 DefinitionThebinomialprobabilitydistributiondefinestheprobabilityofexactlyxsuccessesinntrialsoftheBernoulliprocess Thebernoullitrialisanexperimentwithtwo andonlytwo possibleoucomes 5 1 2MainContent 2 Characteristic onlytwopossibleoutcomes afixnumberofn independenteachother constantp ifpissuccess soq 1 pisfailureandp q 1 5 1 2MainContent 3 FormularWhere n randomsamplesize x numberofsuccesses n x numberoffailures p probabilityofasuccess q 1 p probabilityofafailure 3 Formular exercise AstudybytheInternationalCoffeeAssociationfoundthat52 oftheU S populationaged10andoverdrinkcoffee Forarandomlyselectedgroupof4individuals whatistheprobabilitythat3ofthemarecoffeedrinkers NumberProportionCoffeedrinkers x 3 52Noncoffeedrinkers1 48Totals41 00So p 0 52 1 p 0 48 x 3 n x 1 3 Formular exercise Tosolvetheproblem wesubstitute 5 1 2MainContent 4 UsingthebinomialdistributiontableAnExampleWorkedwithTables AnExampleWorkedwithTables Problem Accordingtoacorporateassociation 50 0 ofthepopulationofVermontwereboatingparticipantsduringthemostrecentyear Forarandomlyselectedsampleof20Vermontresidents withx thenumbersampledwhowereboatingparticipantsthatyear determine a E x b P x 8 c P x 10 d P x 12 e P 7 x 13 Answers a E x np 20 x0 50 10b P x 8 0 2517c P x 10 0 1762d P x 12 0 1201e P x 13 0 9423 P x 6 0 0577P 7 x 13 0 9423 0 0577 0 8846 UsingMicrosoftExcel b P x 8 InacellonanExcelworksheet type BINOMDIST 8 20 0 5 true andyouwillseetheanswer 0 2517c P x 10 InacellonanExcelworksheet type BINOMDIST 10 20 0 5 false andyouwillseetheanswer 0 1762d P x 12 InacellonanExcelworksheet type BINOMDIST 12 20 0 5 false andyouwillseetheanswer 0 1201e P 7 x 13 InacellonanExcelworksheet type BINOMDIST 13 20 0 5 true BINOMDIST 6 20 0 5 true andyouwillseetheanswer 0 8846 5 1 2MainContent 5 MeanandstandarddefinitionTheprobabilitydistributionforadiscreterandomvariabledefinestheprobabilityofadiscretevaluex Mean E x Variance s2 E x 2 BinominaldistributionMeanE x npVariances2 npq 5 1 3Terminologies DiscreterandomvariableBinomialprobabilitydistributionContinuousrandomvariableCountingruleforcombinationsBinomialformula 5 1 3Terminologies ExpectedvalueofadiscreterandomvariableExpectedvalueofabinomialdistributionStandarddeviationofadiscreterandomvariableStandarddeviationofthebinomialdistribution 5 1 4Afterschoolextend 1 catalogsales2 Usbiointroduction 5 2OtherDiscreteProbabilityDistributions 5 2 1Vocabularies词汇5 2 2MainContent主要内容5 2 3Terminologies专业词汇5 2 4Afterschoolextend课外扩展 5 2 1Vocabularies 1 VermontAstateofthenortheastUnitedStatesborderingonCanada Itwasadmittedasthe14thstatein1791 ExploredbySamueldeChamplainin1609 theregionwasfirstpermanentlysettledbytheBritishin1724 ClaimstotheareawererelinquishedbyMassachusettsin1781 NewHampshirein1782 andNewYorkin1790 MontpelieristhecapitalandBurlingtonthelargestcity Population 564 964 佛蒙特州 美国东北部的一个州 与加拿大接壤 它于1791年被接受为美国的第十四个州 1609年塞缪尔 德 查普兰勘探了这个地区 1724年英国人在该地区永远定居了下来 马萨诸塞州 新罕布什尔州和纽约州于1781年 1782年和1790年相继放弃了对这个地区的权利 该州首府为蒙彼利埃 伯林顿是最大的城市 人口564 964 5 2 1Vocabularies 2 downsizevt Tomakeinasmallersize减小 汽车的尺寸 重量和耗油量 carsthatweredownsizedduringthegasoline conscious1970 s 在汽油短缺的七十年代以较小尺寸制造的小汽车acorporationthatdownsizeditspersonnelinresponsetoapooreconomy 因经济不景气而裁员的股份有限公司 5 2 1Vocabularies 3 laylayaside Togiveup abandon 放弃 抛弃 Eg layasideallhopeofrescue 放弃援救的希望layoff Toterminatetheemploymentof aworker especiallytemporarily 暂时解雇员工 停止雇用 工人 尤指暂时性的layon Toapply something byorasifbyspreading it ontoaflatsurface 带有 将某物抹在或好象涂在一平整表面上 Eg laidonathickSouthernaccent 带有很重的南方口音 5 2 1Vocabularies chuckhole 马路上的坑或凹处 segment 段 节 片断 分割 strech 伸展 一段路程 时间 Vermont 佛蒙特州 acme 顶点 极致 vendingmachine 自动售货机 自动贩卖机 printer 印刷工 打印机 reinspect 再考虑 重新考察 glance 扫视 匆匆一看 Hewlett packardLaserJetprinter惠普激光打印机 5 2 2MainContent 1 thePoissondistribution 1 DefinitionThePoissondistributiondefinestheprobabilitythataneventwilloccurexactlyxtimesoveragivenspanoftime space ordistance Thepossiblecountsaretheintegers 5 2 2MainContent 1 thePoissondistribution 2 Assumptions istheaveragenumberofsuccessinasegmentofunitsize theprobabilityofxsuccessinasegmentisthesameforallsegmentsofthesamesize whathappensinonesegmenthasnoinfluenceonanynonoverlappingsegment weimaginedividingtimeorspaceintotinysubsegment Thenthechanceofmorethanonesuccessinasubsegmentisnegligibleandthechanceofexactlyonesuccessinatinysubsegementoflengthtis 5 2 2MainContent 1 thePoissondistribution 3 FormularWhere t sizeofthesegmentofinterestx numberofsuccessesinthesegmentofinterest expectednumberofsuccessinssegmentofunitsizee baseofthenaturallogarithmsystem 2 71828 5 2 2MainContent ThePoissondistributionexample Duringthe12p m 1p m noonhour arrivalsatacurbside 路旁的 bankingmachinehavebeenfoundtobePoissondistributedwithameanof1 3personsperminute Ifx numberofarrivalsperminute determine a E x l 1 3b 5 2 2MainContent c d P x 2 0 8571frompoissonTable l 1 3 k 2IfuseexcelP x 0 InacellinanExcelspreadsheet type POISSON 0 1 3 false P x 1 type POISSON 1 1 3 false P x 2 type POISSON 2 1 3 true 5 2 2MainContent 1 thePoissondistribution 4 MeanandstandarddefinitionE x Var x 5 2 2MainContent 1 thePoissondistribution练习 5 11和5 12 思考 二项分布和泊松分布区别和联系 共同点 theybothrequirethatthetrialsbeindependent replacementorn N 5 5 2 2MainContent 2 thehypergeometricdistribution 1 BackgroundWhenthesampleislargerelativetothesizeofthepopulation thehypergeometricdistributionisthecorrectdistributionforcomputingprobabilitiesfortherandomvariableofinterest thehypergeometricdistributionisformedbytheratioofthenumberofwaysaneventofinterestcanoccuroverthetotalnumberofwaysanyeventcanoccur Case LindellCorporation 5 2 2MainContent 2 thehypergeometricdistribution 2 formulawhere N populationsizeX numberofsuccessesinthepopulationn samplesizex numberofsuccessesinthesamplen x numberoffailuresinthesample 5 2 2MainContent 2 thehypergeometricdistribution 3 example5 1 5 2 3Terminologies Poissondistributionthehypergeometricdistribution twopossibleoutcomepertrial standarddeviationofthepoissondistributionPoissonprobabilitydistributionArithmeticHypergeometricdistribution 5 2 4Afterschoolextend 1 ThePoissondistributionasalimitingformoftheBinomial 翻译 ItshouldbeevidentfromthethreeprinciplesofthePoissonprocessthatthePoissondistributionrelatestothebinomialdistribution AlthoughthePoissonusuallyfindsapplicationsinspaceandtimeproblemsasillustratedbyexamples5 19and5 20 itcanbeviewedasalimitingformofthebinomialdistribution Inthecaseofthebinomial ifnisquitelargeandpissmall theconditionsbegintosimulatethecontinuousspaceortimeregionimplicationsofthePoissonprocess 5 2 4Afterschoolextend TheindependenceamongBernoullitrialsinthebinomialcaseisconsistentwithproperty2ofthePoissonprocess Allowingtheparameterptobeclosetozerorelatestoproperty3 Indeed weshallnowderivethePoissondistributionasalimitingformofthebinomialdistributionwhen andnpremainsconstant Hence ifnislargeandpiscloseto0 thePoissondistributioncanbeused with toapproximatebinomialprobabilities Ifpiscloseto1 wecanstillusethePoissondistributiontoapproximatebinomialprobabilitiesbyinterchangingwhatwehavedefinedtobeasuccessandafailure therebychangingptoavaluecloseto0 5 2 4Afterschoolextend 2 关于shipping相关材料补充Dealersinthecommodityexchangescaneitherbuyforimmediatedelivery spot or cash orforwarddeliveryin say threemonths time Forwarddeliveryisknownasafuturecontract Inthesimplestcase whenpricearerising youcanbuyforimmediatedeliveryandsellforward then providedthecostofwarehouse profit ofcourse thisisnotpossiblewhenpricesarefalling 5 2 4Afterschoolextend 1 dealer 经销商 通常与厂商有密切联系 也可指证券 外汇及黄金市场上的交易商 交易人 经纪人等 speculativedealer交易所投机商 以某种方式待人的人 Eg adoubledealer表里不一的人 两面派 afair plain dealer行为正直的人 2 commodityexchanges商品交易 所 3 forwarddelivery 外贸 远期交货 shortdelivery的对称 Eg Bothcashandforwardoilshowedlargelossesinthemarketbytheendofmorning 早市结束时现货和期货的石油价格下跌很多 注 forward还可做运输 运送 寄送解 Eg Thegoodswillbeforwardedtoyouinthenextweek 货物将于下周发运给你方 5 2 4Afterschoolextend 4 warehouse仓库 货栈 批发商店 大零售商店 warehousing仓库费 仓库储存 5 shipn 船 v 装船 装运 Eg Weexpecttoshipthegoodsnextweek 6 shipment theactofputtinggoodsonaship装运 Eg Portofshipment装运港 goodsbeingcariedbyship船货 到货 Eg ourcustomersaresatisfiedwithyourlastshipment 7 shippingn 海运Eg Shippinginstruction装船要求 shippingadvice装船通知 shippingdocuments装船单据 shippingspace舱位 8 shippern 发货人 5 3TheNormalProbabilityDistribution 5 3 1Vocabularies词汇5 3 2MainContent主要内容5 3 3Terminologies专业词汇5 3 4Afterschoolextend课外扩展 5 3 1Vocabularies 1 highlightn 1 Anareaoraspotinadrawing painting orphotographthatisstronglyilluminated 最亮点或最高区 素描 油画或照片中得到最大量光照的一点或一片 2 Anespeciallysignificantorinterestingdetailorevent 最重要的或最有趣的细节或事件v 1 Tomakeprominent emphasize 使 显得重要 强调的 2 Tobeahighlightof 为 中最突出的事物 3 Tomark importantpassagesoftext withausuallyfluorescentmarkerasameansofmemoryretentionorforlaterreference 标示记号 用荧光笔标出 文本中重要的段落 以便帮助记忆或日后查找参考 5 3 1Vocabularies 2 Midland 1 AcityofcentralMichiganwestofBayCity Chemicalindustriesareimportanttoitseconomy Population 38 053 米德兰 美国密歇根州中部一城市 位于湾市以西 化学工业为其经济的主要支柱 人口38 053 2 Acityofwest centralTexaswest southwestofAbilene Itisanindustrialcenterinacattle ranchingarea Population 89 443 米德兰 美国得克萨斯州中西部一城市 位于艾伯林西南偏西 为该畜牧业区中的工业中心 人口89 443 5 3 1Vocabularies 3 enzymeAnyofnumerousproteinsorconjugatedproteinsproducedbylivingorganismsandfunctioningasbiochemicalcatalysts 酶 由活的有机体产生的各种蛋白质或蛋白质结合体的一种 起生物催化剂的作用4 giftcertificate 礼券 n Acertificateusuallypresentedasagiftthatentitlestherecipienttoselectmerchandiseofanindicatedcashvalueatacommercialestablishment 赠券 赠品证明书 作为赠送礼物的证明书 表明领受人有权在指定的金额范围内在一商场选择商品的证明书 5 3 2MainContent 1 characteristics 1 Animportantfamilyofcontinuousdistributions 2 itisunimodal thatis thenormaldistributionpeaksatasinglevalue 3 symmetrical thisismeansthatthetwoareasunderthecurvebetweenthemeanandanytwopointsequidistantontheeithersideofthemeanareidentical Onesideofthedirtributionisthemirrorimageoftheotherside 4 mean median modeareequal 5 asymptotictotheX axis 6 variationdeterminesthewidthofthenormaldistribution 5 3 2MainContent 2 NormalProbabilityDensityFunctionwhere x anyvalueofthecontinuousrandomvariable populationmean 位置参数 populationstandarddeviation 形状参数 3 14159e baseofthenaturallog 2 71828Note 1 Theareaunderaprobabilitydensityfunctionbetweentwobounds aandb istheprobabilitythatavaluewilloccurwithintheboundedintervalbetweenaandb 2 thetotalareaundereachnormalcurveequals1 5 3 2MainContent 3 theStandardNormalDistribution 5 3 2MainContent 4 Usethestandardnormaltabletofind 1 Thez scoresuchthattheareafromthemidpointtozis0 20 Answer z 0 52 2 TheprobabilityassociatedwithZ P 0 z 1 32 Answer 0 4066 3 Theprobabilityassociatedwithz P 1 10 z 1 32 Answer 0 3643 0 4066 0 7709 4 Theprobabilityassociatedwithz P 1 00 z 1 32 Answer 0 4066 0 3413 0 0653 5 3 2MainContent 5 CalculateStandardNormalDistribution AnExampleIthasbeenreportedthattheaveragehotelcheck intime fromcurbsidetodeliveryofbagsintotheroom is12 1minutes Maryhasjustleftthecabthatbroughthertoherhotel Assuminganormaldistributionwithastandarddeviationof2 0minutes whatistheprobabilitythatthetimerequiredforMaryandherbagstogettotheroomwillbe 1 greaterthan14 1minutes 2 lessthan8 1minutes 3 between10 1and14 1minutes 4 between10 1and16 1minutes AnExample cont Givenintheproblem 12 1minutes s 2 0minutes1 Greaterthan14 1minutesP x 14 1 P z 1 00 5 3413 0 15872 Lessthan8 1minutesP x 8 1 P z 2 00 5 4772 0 0228 Example UsingMicrosoftExcel Problem WhatistheprobabilitythatthetimerequiredforMaryandherbagstogettotheroomwillbe 3 between10 1and14 1minutes InacellonanExcelworksheet typeallononeline NORMDIST 14 1 12 1 2 true NORMDIST 10 1 12 1 2 true andyouwillseetheanswer 0 68264 between10 1and16 1minutes InacellonanExcelworksheet typeallononeline NORMDIST 16 1 12 1 2 true NORMDIST 10 1 12 1 2 true andyouwillseetheanswer 0 8185 5 3 2MainContent 6 approximateareasunderthenormalcurveTherearesomegeneralrules 1 enclosesabout68 ofX s 2 enclosesabout95 ofX s 3 enclosesabout99 ofX s 5 3 2MainContent 7 二项分布的正态近似BinomialProbabilityFunction当实验次数很大时 用手工计算或计算器计算求解二项概率函数是困难的 当试验次数大于20 np 5和n 1 p 5时 正态分布可以作为二项分布的近似 Example假定历史经验表明 某公司发票出错的概率为10 现计算选取100张发票 恰好有12张发票有错的概率 13张或更少张发票出错的概率 计算100次试验中恰好有12次成功的二项概率 采用正态近似计算概率 5 3 3Terminologies 5 3 4Afterschoolextend 1 aboutnormaldistribution 翻译 Themostimportantcontinuousprobabilitiesdistributionintheentirefieldofstatisticsisthenormaldistribution Itsgraph calledthenormalcurve isthebell shapedcurveoffigure6 2 whichdescribesapproximatelymanyphenomenathatoccurinnature industry andresearch Physicalmeasurementsinareassuchasmeteorologicalexperiments rainfallstudies andmeasurementsofmanufacturedpartsareoftenmorethanadequatelyexplainedwithanormaldistribution 5 3 4Afterschoolextend 1 aboutnormaldistribution 接前页 Inaddition errorsinscientificmeasurementsareextremelywellapproximatedbyanormaldistribution In1733 AbrahamDemoivredevelopedthemathematicalequationofthenormalcurve Itprovidedabasisforwhichmuchofthetheoryofinductivestatisticsisfounded ThenormaldistributionisoftenreferredtoastheGaussiandistribution inhonorofKarlFriedrichGauss 1777 1855 whoalsoderiveditsequationfromastudyoferrorsinrepeatedmeasurementsofthesamequantity 5 3 4Afterschoolextend 2 normalapproximationtothebinomialProbabilitiesassociatedwithbinomialexperimentsarereadilyobtainablefromtheformulab x n p ofthebinomialdistributionorfromTableA 1whennissmall Inaddition binomialprobabilitiesarereadilyavailableinmanysoftwarecomputerpackages However itisinstructivetolearntherelationshipbetweenthebinomialandnormaldistribution Insection5 6weillustratedhowthePoissondistributioncanbeusedtoapproximatebinomialprobabilitieswhennisquitelargeandpisverycloseto0or1 boththebinomialandPoissondistributionsarediscrete 5 3 4Afterschoolextend 2 normalapproximationtothebinomial 接前页 Thefirstapplicationofacontinuousprobabilitydistributiontoapproximateprobabilitiesoveradiscretesamplespaceisdemonstratedbyexample6 12 wherethenormalcurvewasused Thenormaldistributionisoftenagoodapproximationtoadiscretedistributionwhenthelattertakesonasymmetricbellshape Fromatheoreticalpointofview somedistributionsconvergetothenormalastheirparametersapproachcertainlimits Thenormaldistributionisaconvenientapproximatingdistributionbecausethecumulativedistributionfunctionissoeasilytabled 5 4OtherContinuousProbabilityDistributions 5 4 1Vocabularies词汇5 4 2MainContent主要内容5 4 3Terminologies专业词汇5 4 4Afterschoolextend课外扩展5 4 5Literature文献导读 5 4 1Vocabularies 1 elapseToslipby pass 悄悄过去 消逝 Weekselapsedbeforewecouldstartrenovating 翻译 几周过去后我们才开始修复n 名词 Passage lapse 过去 逝去 metagainafteranelapseofmanyyears 翻译 多年后又相逢 5 4 1Vocabularies 2 drive in 1 n Anestablishmentdesignedtopermitcustomerstoremainintheirmotorvehicleswhilebeingaccommodated 免下车设施 顾客接受服务时可不下车的营业场所或娱乐设施 如免下车餐馆 免下车电影院 美 让顾客不下汽车即可吃饭 办事 看戏等的 路旁餐馆 银行 露天电影院Eg Let seatatthedrive intonight 咱们今晚在 免下车 餐馆吃饭吧 2 adj 免下车的 路边服务式的 饭店 电影院等 露天营业式的 3 drivein赶进 敲进 5 4 1Vocabularies 3 tellern 1 Onewhotells 叙述者 Eg atelleroftalltales 讲述难以令人相信的故事的人 2 Abankemployeewhoreceivesandpaysoutmoney 出纳员 接受和支付钱款的银行雇员 3 Amachine asinabank thatautomaticallyconductspersonalfinancialtransactionsinresponsetoaclient suseofacodedcard 银行柜员机 银行使用的能对委托人持用的有密码的卡片作出反应并自动进行个人财产交易的机器 4 Apersonappointedtocountvotesinalegislativeassembly 计票人 被指派在议会中数票的人 5 4 1Vocabularies