(随机抽样)

Randomsampling(随机抽样)•Independent（独立性）•Representative（代表性）–ThepoliticalsystemoftheUSTheUSSenate(参议院）•EachstateallowstwosenatorsSothereareabout100senatorsin50states.TheHouseofrepresentatives（众议院）•Thenumberofrepresentativesfromeachstatedependsonitspopulation–California56–NewYork33–Texas32–Delaware3–Tennessee12–Florida25Usingrandomnumberstoimplementrandomsampling128934567…100…SamplingprobabilitySussexUniversityhas1000students.200ofthemareforeignstudents,othersareAmerican.Whatistheprobabilityofrandomlysamplingaforeignstudentinonetrial?Thesamplingsize•Ifyouselect1studentfrom1000,thentheoutcomeofprobabilitywouldbeeither0%or100%.•Ifyouselectmany,thentheoutcomewouldapproach20%Thecoin-tossexperimentExperimenterTimesoftoss,nHeaduptimes,mFrequency,m/nBuffon404020480.5069Pearson1200060190.5016Pearson24000120120.5005Head/TailThelawoflargenumbers(大数定律)Whensamplingsizeissufficientlylarge,thesamplingmeanoutcomecalculatedfromrepeatedobservationsofarandomvariablemustapproachthedistributionprobability.limP{|f/n–p|ε}=1n∞f:frequencyoftheeventsp:theoreticalprobabilityn:samplingsizeε:aninfinitesimalnumberThelawoflargenumbersThelawoflargenumbersisthefoundationuponwhichsuchbusinessenterprisesasgamblingcasinosandinsurancecompaniesarebuiltSamplingprobabilitySussexUniversityhas1000students.200ofthemareforeignstudents,othersareAmerican.Whatistheprobabilityofgetting2foreignstudentintwosamples?FFAAAFFASamplingpoint(样本点）Samplingspace（样本空间）{FF,FA,AA}{p2,2pq,q2}p=P[F]=200/1000q=P[A]=800/1000p+q=1Theprobabilityofindependentlysampling2students{FFF,FFA,FAA,AAA}{p3,3p2q,3pq2,q3}Theprobabilityofindependentlysampling3studentsGivenp=0.2,q=0.8,thentheprobabilityvalues:{0.008,0.096,0.384,0.512}Probabilitydistribution(3samples)00.20.40.6NumbersofAmericanStudentsP0123ProbabilitydistributionForsamplesof1,p+qForsamplesof2,(p+q)2=p2+2pq+q2Forsamplesof3,(p+q)3=p3+3p2q+3pq2+q3…Forsamplesofn,(p+q)n=ThecoefficientsoftheexpandedtermsPascal’striangle:1112113311464115101051n12345…Forsamplesof4,(p+q)4=p4+4p3q+6p2q2+4pq3+q4Thebinomialdistribution(二项分布)Pn(k)=Cnkpkqn-kX012…k…nPqnnpqn-1Cn2p2qn-2…Cnkpkqn-k…pnDistributionseries(分布列）)!(!!nNnNnNCnNCombinationsruleAsampleofnelementstobechosenfromasetofNelements.Thenthenumberofdifferentsamplesofnelements:Characteristicsofthebinomialdistribution•Thereareonlytwopossibleoutcomesoneachtrial,ForA•Thetrialsareindependent•TheprobabilityofF(foraforeignstudent)remainsthesamefromtrialtotrial,pThebinomialdistribution(二项分布)Pn(k)=Cnkpkqn-kX012…k…nPqnnpqn-1Cn2p2qn-2…Cnkpkqn-k…pnDistributionseries(分布列）Property1.0≤Pn(k)≤1Property2.∑Pn(k)=1k=1n(p+q)n=1BasicprobabilityrulesRule1.AnyprobabilityP(A)isanumberbetween0and1,thatis0≤P(A)≤1.Rule2.ThecollectionSofallpossibleoutcomeshasprobability1,thatisP(S)=1.AdiscreterandomvariableXcanassumefivepossiblevalues:2,3,5,8,10.Itsprobabilitydistributionisshownhere:X235810p(X)0.150.100.250.25a.Whatisp(5)?b.WhatistheprobabilitythatXequals2or10c.Whatisp(X8)?Themean,expectedvalue,orexpectationofadiscretevariablexisdefinedasμ=E(x)=∑xp(x)WhatistheaveragesizeofanAmericanfamily?HereisthedistributionofthesizeofAmericanfamiliesaccordingtoCensusBureaustudies.Personsinfamily2345678Fractionoffamilies.396.231.212.096.038.017.008Randomdiscretevariables(离散型随机变量)•Thelawoflargenumbers•Thebinomialdistribution甲、已两台车床加工同一型号产品，生产1000只所含次品数各用X,Y表示，经过一端时间观察，X,Y的分布列各为X0123P0.70.10.10.1Y0123P0.50.30.20Whichmachinehasabetterquality?Homework1.Page45,question3.62.Pleasewriteoutamacroprogramforgenerating200randomnumbersintherangefrom0to100andstoringthedatainthefirsttwocolumnsonExcelspreadsheet.3.Inhumanbeingsthesexratioofnewborninfantsisabout100maleto105female.Werewetotake10,000randomsamplesof6newborninfantsfromthetotalpopulationofsuchinfantsforoneyear,whatwouldbetheexpectedfrequencyofgroupsof6males,5males,4males,andsoon?