概率統(tǒng)計中文概統(tǒng)課件lecture

Lecture17ThetDistributionConfidenceIntervalsformeanThetDistributionConsidertwoindependentrandomvariablesYandZ,suchthatZhasastandardnormaldistributionandYhasadistributionwithndegreesoffreedom.SupposearandomvariableXisdefinedby

ThenthedistributionofXiscalledthetdistributionwithndegreesoffreedom.Thep.d.f.LetW=Y,thenSinceRelationtotheNormalDistributionWhen,g(x)convergestothep.d.f.f(x)ofthestandardnormaldistributionforeveryvalueofx.Whennislarge,thetdistributionwithndegreesoffreedomcanbeapproximatedbythestandardnormaldistribution.n=1(Cauchy)n=2n=5n=10(normal)MeanandVarianceof

thetDistributionThemeandoesnotexistwhenn=1.Itexistsandisequalto0foranyvalueofn>1.IfXhasatdistributionwithndegreesoffreedom(n>2),thenVar(X)=n/(n-2).RelationtoRandomSamplesfrom

aNormalDistributionSupposethatX1,…,Xnformarandomsamplefromanormaldistributionwithmeanmandvariance.Since

andtheyareindependentofeachother,Define Theyareoftenreferredtoassamplevarianceandsamplestandarddeviation.Whenthevarianceisknown,

whenitisunknown,wecanreplacebyS, Theycanbeusedtomakestatisticalinference aboutm.9小小的復習：很多時候，我們通過樣本來了解總體?？傮w樣本10基本概念總體與樣本總體：所有個體的全體樣本：觀測到的個體為什么需要抽樣？1）總體無法得到。

例：光臨麥當勞的所有顧客（無限總體）。

2）時間和成本不允許。

例：美國總統(tǒng)選舉的民意測驗。 3）實驗具有破壞性。

例：測量產(chǎn)品的壽命。統(tǒng)計分析的任務:通過樣本的統(tǒng)計量來了解總體的參數(shù)。參數(shù)與統(tǒng)計量參數(shù)：關(guān)于總體的度量，如：，p，統(tǒng)計量：關(guān)于樣本的度量，如：樣本統(tǒng)計量總體參數(shù)參數(shù)的點估計14考慮到所有可能的樣本...

15估計的誤差樣本均值的標準誤差樣本均值的區(qū)間估計樣本均值僅僅是未知參數(shù)的一個近似值，它沒有反映出這個近似值的誤差范圍，使用起來把握不大。一個典型的區(qū)間估計就是：

點估計±誤差范圍。ConfidenceIntervalsConfidenceIntervalsforanArbitraryParameterqSupposeX1,…,Xnformarandomsamplefromadistributioninvolvinganunknownparameterq.SupposethattwostatisticsA(X1,…,Xn)andB(X1,…,Xn)canbefoundsuchthat,regardlessofthetruevalueofq,

where(1-)isafixedprobability.IftheobservedvaluesofA(X1,…,Xn)andB(X1,…,Xn)areaandb,thenwesaythattheinterval(a,b)isaconfidenceintervalforqwithconfidencecoefficient(1-),orqliesintheinterval(a,b)withconfidence(1-).ConfidenceIntervalsfortheMeanofaNormalDistributionWhentheVarianceisKnownSupposethattheobservedvaluesx1,…,xncomefromarandomsamplefromanormaldistributionwithunknownmeanmandknownvarianceSotheconfidenceintervalformisExampleContinued:

95%ConfidenceIntervalformWhenisknownItiscalculatedthatTheconfidenceintervalformisLetConsiderintervalshavingtheform

95%ofallvaluesinsidepurpleband,coversinsidepurpleband,coversoutsidepurpleband,missesInterpretationofConfidenceIntervalsBeforeobtainingtheobservedvaluesoroverrepeatedsampling,thereisprobability95%thatthesamplemeanwilllieinthepurpleband,so95%ofallintervalsformedbywillcoverthepopulationmean,orItisnotcorrecttostatethatmliesintheconfidenceintervalwithprobability95%.Wecanonlysaythatweare95%confident.2295%置信區(qū)間的含義抽取100個樣本，計算出100個樣本的平均值和100個這樣的區(qū)間，那么，它們當中大約有95個包含了未知的總體均值。因此，可以以95%的把握(置信度)相信落在一個這樣的區(qū)間里面。一旦你看到“95%信心”等字眼，表明這個研究結(jié)果雖不能說百分之百確定，但已相當有把握。

ConfidenceIntervalsfortheMeanofaNormalDistributionWhentheVarianceisUnknownSupposethatX1,…,Xn

formarandomsamplefromanormaldistributionwithunknownmeanmandunknownvariance.:theaupper-quantileoftn-1distribution.Usuallytheconfidencecoefficient1-

a=0.9,0.95,0.99,etc.TheconfidenceintervalformisExampleContinuedItiscalculatedthat

Sothe90%confidenceintervalestimateofthepopulationmeanis案例：商店選址客流量張先生是某集團的企劃部經(jīng)理，在今年的規(guī)劃中，集團準備在某地新建一家新的零售商店。張先生目前正在做這方面的準備工作。其中一項便是進行市場調(diào)查。在眾多的信息中，經(jīng)過該地行人數(shù)量是要考慮的一個很重的方面。張先生委托人進行了兩個星期的觀察，得到經(jīng)過該地人數(shù)如下：