概率統(tǒng)計中文概統(tǒng)課件lecture

Introductionto

ProbabilityandStatistics

Instructor:YundongTu(涂云東) Phone:62760219 Email: Office: Room475, GuanghuaBuildingNo.2

助教郇鈺，劉進(jìn)，謝雨辰，課程郵箱：賬號：密碼：gsm2016權(quán)限：只允許下載課件資料信仰不能取代數(shù)字

-----HenrySpencer

Case1:電視訪問一位能說會道的母親，她的孩子得了白血病，他們住在高壓電線附近。住在高壓電線附近會導(dǎo)致兒童得白血病嗎？在大眾心目中，往往相信軼聞。但我們應(yīng)該要心存疑問癌癥研究所花費(fèi)5年的時間和500萬元為這個問題搜集資料。結(jié)論是：在白血病和暴露在高壓電線所產(chǎn)生的磁場之間，找不到相關(guān)關(guān)系。數(shù)據(jù)比軼聞可靠，因?yàn)閿?shù)據(jù)可以有系統(tǒng)地描繪出整體的情況，而軼聞只聚焦于少數(shù)特例。好的數(shù)據(jù)勝過軼聞。比起軼聞和光大聲嚷嚷預(yù)測未來，數(shù)據(jù)要客觀得多。數(shù)據(jù)勝過自封的專家

Case2:對于便利連鎖店（如7-11），店員的熱情招呼和微笑服務(wù)會提高顧客滿意？深入的數(shù)據(jù)分析證明這個想法完全錯誤的，彬彬有禮的店員反而降低了顧客滿意。可能的原因?Case3:中國人工作辛苦，工作時間長，工資收入低是有目共睹的事實(shí)。中國人勤勞是全世界公認(rèn)的美德。那么中國人的勤勞能換來企業(yè)的高效率嗎？行業(yè)世界500強(qiáng)排名公司名稱總部所在地營業(yè)額（百萬美元）員工數(shù)量人均營業(yè)額（百萬美元）煉油1荷蘭皇家殼牌石油公司荷蘭484489.090,0005.382埃克森美孚美國452926.099,1004.574英國石油公司英國386463.083,4004.635中國石油化工集團(tuán)公司中國375214.01,021,9790.376中國石油天然氣集團(tuán)公司中國352338.01,668,0720.212012年《財富》世界500強(qiáng)企業(yè)將2012年《財富》世界500強(qiáng)企業(yè)重新排序。排名首位的荷蘭，共有12家企業(yè)，平均人均營業(yè)額13.8百萬美元；中國進(jìn)入世界500強(qiáng)企業(yè)共有79家，平均人均營業(yè)額0.6百萬美元，排名第二十六。差距為23倍。或許在過去，公司還能拍腦門做決策，無視正確的統(tǒng)計數(shù)據(jù)，因?yàn)樗麄冞€能靠跟隨行業(yè)“慣例”糊弄過去。如今，有效的利用統(tǒng)計數(shù)據(jù)是競爭的必要條件。更為重要的是，在你的競爭對手之前找到并充分利用統(tǒng)計數(shù)據(jù)，將成為打造你的競爭優(yōu)勢的關(guān)鍵所在。

如今的中國企業(yè)的管理正由粗放型轉(zhuǎn)向精細(xì)型管理。

統(tǒng)計分析能夠帶給你價值，對數(shù)據(jù)有敬畏之心的，率先進(jìn)行投入的企業(yè)，將更有機(jī)會在未來競爭中取得先機(jī)。

----《哈佛商業(yè)評論》ProbExample:Let’sMakeaDealMontyHallProblemSupposeyou'reonagameshow,andyou'regiventhechoiceofthreedoors:Behindonedoorisacar;behindtheothers,goats.Youpickadoor,sayNo.

1[butthedoorisnotopened],andthehost,whoknowswhat'sbehindthedoors,opensanotherdoor,sayNo.

3,whichhasagoat.Hethensaystoyou,"DoyouwanttopickdoorNo.

2?"Isittoyouradvantagetoswitchyourchoice?Solution:ConditionalProbabilityandBayes’Formula

StatExample:LinearRegressionTextbook ProbabilityandStatistics,RevisedEditionbyXiangzhongFang,LigangLu,DongfengLi,HigherEducationPress,2005.LectureNotesReferences陳希孺(2000)，概率論與數(shù)理統(tǒng)計，中國科學(xué)技術(shù)大學(xué)出版社。課堂要求不遲到，不早退上課積極參與討論主動關(guān)閉手機(jī)、筆記本電腦等電子產(chǎn)品CourseAssessmentYourfinalgradewillbebasedonfivecomponents:Homeworkassignedonaweeklybasiswillaccountfor25%.Youmaydiscusshomeworkproblemswithotherstudents,butyoumustwritethemupindependently.Homeworkisdueatthebeginningofclassontheduedate.Notethathomeworkthatisturnedinlaterthantheendofclassontheduedatewillnotbegraded.Itisunderstoodthatfromtimetotimeyourschedulemaynotallowyoutoturninyourhomeworkontime,soyourlowesthomeworkscorewillbedroppedwhencomputingyourfinalgrade.Midtermexam(Apr13,2015)willaccountfor25%.Finalexam(Jun15,2016)willaccountfor40%.Smallprojectwillaccountfor10%.Bonusquizzes(randomlyassigned),eachaccountingfor2%Notethatcheatingisstrictlynotallowedandwillresultinzeroscoreontherespectivepart.IntroductionofProbabilityExperiment(試驗(yàn))Supposeacoinistossedonceandtheupfaceisrecorded.Theresultweseeandrecordiscalledanobservation,ormeasurement,andtheprocessofmakinganobservationiscalledanexperiment.Thepointisthatastatisticalexperimentcanbealmostanyactofobservationaslongastheeisuncertain.Definition:Anexperimentisanactorprocessofobservationthatleadstoasingleethatcannotbepredictedwithcertainty.

E.g.,Considertheexperimentofcountingthenumberofcustomersatarestaurantonaparticularday.Thebasicpossibleesofthisexperimentare0,1,2,3,…

ExperimentThefeaturesofanexperimentare:eachofitspossibleescanbespecifiedbeforetheexperimentisperformed;oneandonlyoneofthepossibleexperimentaleswilloccur;thereisuncertaintyassociatedwithwhichonewilloccur.

ExperimentSamplePoint,SampleSpace,EventAbasicpossibleeofanexperimentiscalledasamplepoint(樣本點(diǎn)).Thesamplespace(樣本空間)foranexperimentisthesetofallsamplepoints.Anevent(事件)isaspecificcollectionofsamplepoints.

ExperimentSampleSpaceEventThenumberofcustomersatarestaurantonaparticularday{0,1,2,…}Thenumberofcustomersisatleast100Selectapartforinspection{Defective,Nondefective}DefectiveTheheightofarandomlychosenPKUstudent(cm)[120,220]Theheightisnolessthan175cmAgiveneventissaidtohaveoccurrediftheeoftheexperimentisoneoftheesintheevent.E.g.,拋骰子，S={1,2,3,4,5,6}。EventA:拋出的是偶數(shù)，即A={2,4,6}。當(dāng)觀察到s=2,我們就說事件A發(fā)生了。ProbabilityasNumericalMeasureofUncertaintyProbabilityisanumericalmeasureofthelikelihoodthataneventwilloccur.Probabilitiescouldbeusedasmeasuresofthedegreeofuncertainty.“如果我有當(dāng)國王的機(jī)遇，那么我就有戴上皇冠的命”

-----麥克白《莎士比亞》機(jī)遇的數(shù)學(xué)叫做概率。我們有多少能耐可以了解它。概率相對頻率（客觀概率）：

多次重復(fù)做一個實(shí)驗(yàn)時，那么其概率就是事件最終發(fā)生的次數(shù)比率。個別概率（主觀概率）：生活中大部分事件不會重演，個別概率是個人對某種結(jié)果發(fā)生的主觀估計。

骰子可以一擲再擲，而我們的人生呢？TheKP&LProblemKP&Lcompanyisstartingaprojectdesignedtoincreasethegeneratingcapacityofoneofitsplants.Theprojectisdividedintotwosequentialstages:stage1(design)andstage2(construction).Managementcannotpredictbeforehandtheexacttimerequiredtocompleteeachstageoftheproject.Ananalysisofsimilarconstructionprojectshasshowncompletiontimesforthedesignstageof2,3,or4monthsandcompletiontimesfortheconstructionstageof6,7,or8months.Becauseofthecriticalneedforadditionalelectricalpower,managementhassetagoalof10monthsforthecompletionoftheentireproject,andmanagementwantstoknowtheprobabilitythattheprojectwillbefinishedontime.SampleSpacefortheKP&LProblemThereare(3)(3)=9esinthesamplespace.CompletionTime(months)Stage1 Stage2 TotalProject(Design)(Construction)ExperimentaleCompletionTime 2 6 (2,6) 8 2 7 (2,7) 9 2 8 (2,8) 10 3 6 (3,6) 9 3 7 (3,7) 10 3 8 (3,8) 11 4 6 (4,6) 10 4 7 (4,7) 11 4 8 (4,8) 12ProbabilitiesProbabilitycanbeinterpretedastherelativefrequencyoftheoccurrenceofaneventiftheexperimentisrepeatedalargenumberoftimes.TheKP&Lproblem:CompletionTime(months) NumberofPastProjectsWithStage1Stage2eTotalTimeTheseCompletionTimes Probability 2 6 (2,6) 8 6 6/40=.15 2 7 (2,7)9 6 6/40=.15 2 8 (2,8)10 2 2/40=.05 3 6 (3,6) 9 4 4/40=.10 3 7 (3,7)10 8 8/40=.20 3 8 (3,8)11 2 2/40=.05 4 6 (4,6)10 2 2/40=.05 4 7 (4,7)11 4 4/40=.10 4 8 (4,8)12 6 6/40=.15 Total 40 Total1.00Theprobabilityoftheprojectbeingfinishedontimeis.15+.15+.05+.10+.20+.05=.701.4SetTheoryProbabilityTheorySetTheorysamplespaceofanexperimente.g.ThenumberofcustomersatarestaurantonaparticulardaythesetofallpossibleesSS={0,1,2,3,…}ane(asamplepoint)e.g.100customersanelementinthesetS100anevente.g.atleast100customersasubsetofpossibleesinSA={100,101,102,…}RelationsofSetTheoryAnesisamemberofSEventAiscontainedineventB:everyethatbelongstothesubsetdefiningAalsobelongstothesubsetdefiningBEmptyset:thesubsetthatcontainsnoesOperationsofSetTheoryUnions.TheunionofAandBisdefinedtobetheeventcontainingallesthatbelongtoeitherAorB,ortheeventthateitherAorBwouldoccur.

theunionofneventsisdefinedtobetheeventwhichcontainsallesthatbelongtoatleastoneofthesenevents,ortheeventthatatleastoneoftheseneventswouldoccur.

theunionofaninfinitesequenceofevents.SomeNotationIntersections.TheintersectionofAandBisdefinedtobetheeventcontainingallesthatbelongbothtoAandB,ortheeventthatbothAandBwouldoccur.

theintersectionofneventsisdefinedtobetheeventwhichcontainsallesthatarecommontoallthesenevents,ortheeventthatalltheseneventswouldoccur.

theintersectionofaninfinitesequenceofevents.SomeNotationComplements.ThecomplementofAisdefinedtobetheeventcontainingallesinSthatdonotbelongtoA,ortheeventthateventAwouldnotoccur. e.g.Disjointevents（不相交事件，互斥事件）.AandBaredisjoint,ormutuallyexclusive,iftheyhavenoesincommon.Moregenerally,acollectionofeventsA1,A2,...,Anissaidtobedisjoint/mutuallyexclusiveifnotwoofthemhaveanyesincommon.

1.5DefinitionofProbabilityDefinition:Aprobabilitymeasure,orsimplyaprobability,onasamplespaceSisaspecificationofnumbersPr(A)foralleventsAthatsatisfyAxioms1,2,and3.Comment1933年，蘇聯(lián)大數(shù)學(xué)家Kolmogorov制定了這個概率論的公理體系，其中對概率是什么不加定義，只指出關(guān)于其運(yùn)算所必須遵守的幾條規(guī)則，這樣就回避了如何定義概率這個難題，在這個基礎(chǔ)上，建立起了概率論的宏偉大廈?，F(xiàn)今談概率，不