lecture13-normal distribution讲座正态分布.pptx

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2024-08-12 发布于北京
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lecture13-normal distribution讲座正态分布.pptx

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Normaldistribution1

Normaldistribution2Themostcommonlyuseddistributioninstatistics.Oftenfoundtobeagoodmodelforphysicalvariableslikeweight,height,temperature,voltage,pollutionlevel,andhouseholdesorstudentgrades,etcNormaldistributioniscontinuous

3Thisfigurerepresentsaplotofthenormalprobabilitydensityfunctionwithmean?andstandarddeviation?.Becausethetotalareaunderthedensitycurveisequalto1Thereisacorrespondencebetweenareaandprobability

68-95-99.7%Rule4Notethatthecurveissymmetricabout?,sothat?isthemedianaswellasthemean.About68%ofthepopulationisintheinterval???.About95%ofthepopulationisintheinterval??2?.About99.7%ofthepopulationisintheinterval??3?.

Normaldensitieswithdifferentlocationandscaleparameters.5

6NormalR.V.:

pdf,mean,andvarianceTheprobabilitydensityfunctionofanormalpopulationwithmean?andvariance?2isgivenbyIfX~N(?,?2),thenthemeanandvarianceofXaregivenby

7StandardUnitsTheproportionofanormalpopulationthatiswithinagivennumberofstandarddeviationsofthemeanisthesameforanynormalpopulation.Forthisreason,whendealingwithnormalpopulations,weoftenconvertfromtheunitsinwhichthepopulationitemswereoriginallymeasuredtostandardunits.Standardunitstellhowmanystandarddeviationsanobservationisfromthepopulationmean.

StandardNormaldistribution.8Normaldistributionwithstandardparametersμ=0,?=1AStandardNormalvariable,usuallydenotedbyz-scorex–??z=

Example9Thermometersthataresupposedtogivereadingsof0oCatthefreezingpointofwater.Testsonalargesampleoftheseinstrumentsrevealthatatthefreezingpointofwater,somethermometersgivereadingsbelow0o(denotedbynegativenumbers)andsomegivereadingsabove0o(denotedbypositivenumbers).Assumethatthemeanreadingis0oCandthestandarddeviationofthereadingsis1