概率统计中文概统课件lecture

Lecture12TheLawofLargeNumbersTheNormalDistributionTheCentralLimitTheoremTheLawofLargeNumbersConvergenceinProbability.SupposeZ1,Z2,...isasequenceofrandomvariables.Itissaidthatthissequenceconvergestoagivennumberbinprobabilityifforanygivennumber,

Thisisrepresentedby

ContinuousFunctionsofRandomVariablesIf,andifg(z)isafunctionthatiscontinuousatz=b,thenIfand,andifg(z,y)iscontinuousat(z,y)=(b,c),then4很多时候，我们通过样本均值来了解总体均值。总体均值样本均值Lawoflargenumbers.SupposethatX1,...,Xnformarandomsamplefromadistributionforwhichthemeanis,andletdenotethesamplemean.Then

Proof.Assumethedistributionfromwhichtherandomsampleistakenhasafinitevariance.FromtheChebyshevinequality,forany,Remark.Ifalargerandomsampleistakenfromadistributionforwhichthemeanisunknown,thenthearithmeticaverageofthevaluesinthesamplewillusuallybeacloseestimateoftheunknownmean.TheNormalDistributionThenormaldistributionisthesinglemostimportantprobabilitydistribution.ContinuousDistributionManyreal-lifedatasetslooklikethisone,thenamegiventothisgeneralshapeis“normal”.

NormaldistributionImportanceofNormalDistribution1.DescribesManyRandomProcessesorContinuousPhenomena2.Thecentrallimittheorem.Ifalargerandomsampleistakenfromsomedistribution,theneventhoughthisdistributionisnotitselfapproximatelynormal,manyimportantfunctionsoftheobservationsinthesamplewillhavedistributionswhichareapproximatelynormal.3.BasisforClassicalStatisticalInferenceDefinitionoftheNormalDistributionArandomvariableXhasanormaldistributionwithmeanandvariance ifXhasacontinuousdistributionwithp.d.f.

TheShapeoftheNormalDistributionThep.d.f.ofanormaldistributionissymmetricwithrespecttothepointx=m.

LinearTransformationTheorem.IfXhasanormaldistributionwithmean andvarianceandifY=aX+b,whereaandbaregivenconstantsand,thenYhasanormaldistributionwithmeanandvariance.TheStandardNormalDistributionThenormaldistributionwithmean0andvariance1iscalledthestandardnormaldistribution.Thep.d.f.ofZthatfollowsthestandardnormaldistributionisdenotedbythesymbol,andthed.f.isdenotedbythesymbol.NormalDistributionProbabilityProbabilityis

areaunder

curve!InfiniteNumberofTablesNormaldistributionsdifferbymean&standarddeviation.Eachdistributionwouldrequireitsowntable.That’saninfinitenumber!StandardizetheNormalDistribution

Onetable!Normal

DistributionStandardizedNormalDistributionStandardizingExampleNormal

DistributionStandardizedNormalDistributionNormalProbabilityTablesExample:P(Z<2.00)=.9773

TheStandardizedNormaltableinthetextbook(Appendix)givesthevalueofforZ02.00.9773Noticethat Sovaluesofcanbederivedforz<0.IfarandomvariableXhasanormaldistributionwithmeanandvariance,thenthevariablehasastandardnormaldistribution.Soprobabilitiesforanynormaldistributioncanbederived.NormalDistributionThinkingChallengeYouworkinQualityControlforGE.Lightbulblifehasanormaldistributionwith

=2000

hours&=200

hours.What’stheprobabilitythatabulbwilllastA.between2000

&2400

hours?B.lessthan1470hours?Solutionfor

P(2000

X

2400)Normal

Distribution

.4773StandardizedNormalDistributionSolutionfor

P(X1470)Normal

Distribution.4960

.0040.5000StandardizedNormalDistributionFindingXValuesforKnownProbabilitiesNormalDistributionStandardizedNormalDistribution

.1217

.1217PropertiesoftheNormalDistributionTheareaunderthepartofanormalcurvethatlieswithin1standarddeviationofthemeanisapproximately0.68,or68%;within2standarddeviations,about0.95,or95%;andwithin3standarddeviations,about0.997,or99.7%.LinearCombinationsofNormallyDistributedVariablesTheorem.IftherandomvariablesX1,...,XkareindependentandifXihasanormaldistributionwithmeanandvariance(i=1,...,k),thenthesumX1+...+Xkhasanormaldistributionwithmean andvariance.Corollary1.IftherandomvariablesX1,...,Xkareindependent,ifXihasanormaldistributionwithmeanandvariance(i=1,...,k),andifa1,...,akandbareconstantsforwhichatleastoneofthevaluesa1,...,akisdifferentfrom0,thenthevariablea1X1+...+akXk+bhasanormaldistributionwithmeanandvariance .Corollary2.SupposethattherandomvariablesX1,...,Xnformarandomsamplefromanormaldistributionwithandvariance,andletdenotethesamplemean.Thenhasanormaldistributionwithmeanandvariance.ExampleSupposethattheheights,ininches,ofthewomeninacertainpopulationfollowanormaldistributionwithmean65andstandarddeviation1,andthattheheightsofthemenfollowanormaldistributionwithmean68andstandarddeviation2.Supposethatonewomanisselectedatrandom,andindependently,onemanisselectedatrandom.Whatistheprobabilitythatthewomanwillbetallerthantheman?

Solution:LetWdenotetheheightoftheselectedwoman,andletMdenotetheheightoftheselectedman.ThenthedifferenceW-Mhasanormaldistributionwithmean65-68=-3andvariance Let ThenZhasastandardnormaldistribution.SoExample:DeterminingaSampleSizeSupposethatarandomsampleofsizenistobetakenfromanormaldistributionwithmeanandvariance9.Whatisthemiminumvalueofnforwhich

Solution:Thesamplemeanhasanormaldistributionwithmeanandstandarddeviation

Let,thenZhasastandard normaldistribution,and Thesamplesizemustbeatleast35.33统计分析的任务通过样本的统计量来了解总体的参数。总体参数p样本统计量为什么需要抽样？1）总体无法得到。

例：光临麦当劳的所有顾客（无限总体）。

2）时间和成本不允许。

例：美国总统选举的民意测验。 3）实验具有破坏性。

例：测量产品的寿命。抽取的样本不同，那么算出的平均值也不同

抽样分布抽取的样本不同，那么算出的平均值也不同。需要了解样本平均值的分布，即它的抽样分布。37关于抽样分布的神奇现象对于简单随机抽样不管总体的分布是什么形态，设它的均值是，方差存在，是2。只要样本的容量n很大，那么样本的均值总是近似服从正态分布(中心极限定理)

Ifalargerandomsampleistakenfromanydistributionwithmeanandvariance, regardlessofthedistributionalform,Thedistributionofthesumwillbeapproximatelyanormaldistributionwithmean andvariance.TheCentralLimitTheoremExample:TossingaCoinSupposeafaircoinistossed900times.Whatistheprobabilityofobtainingmorethan495heads?Fori=1,...,900,letXi=1ifaheadisobtainedontheithtossandletXi=0otherwise.ThenE(Xi)=1/2andVar(Xi)=1/4.Fromthecentrallimittheorem,thetotalnumberofheads