Normal distribution

《Normal distribution》由会员上传分享，免费在线阅读，更多相关内容在学术论文-天天文库。

1、Normaldistribution/GaussiandistributionSumarizeProbabilitydensityfunctionTheredcurveisthestandardnormaldistributionCumulativedistributionfunctionNotationμ∈R—meanParameters2σ>0—varianceSupport（定义域）x∈R(-∞,+∞)pdf(概率密度函数)CDF（累计分布函数）MeanμMedianμMode（众数）μVarianceSkewness（偏度）01Kurtosis（峰度）3Standar

2、dnormaldistributionIfμ=0andσ=1,thedistributioniscalledthestandardnormaldistributionortheunitnormaldistribution,andarandomvariablewiththatdistributionisastandardnormaldeviate.21关于偏度为0和峰度为3的原因见下文Moments部分2积分和为1的证明：先令I=∫f(x)dx，再将一重积分转换为二重积分，再通过极坐标变换得出Properties·IfX~N(μ,),foranyrealnumbersaandb

3、,then:aX+b~N(aμ+b,2a).·IfX~N(μX,X)andY~N(μY,Y)aretwoindependentnormalrandomvariables,then：3U=X+Y~N(μX+μY,X+Y).4V=X-Y~N(μX-μY,X+Y).·IfXandYarejointlynormalanduncorrelated,thentheyare5independent.MomentsIfX(meanis0)hasanormaldistribution,thesemomentsexistandarefiniteforanypwhoserealpartisgrea

4、terthan−1.Foranynon-negativeintegerp,theplaincentralmomentsare6Heren!!denotesthedoublefactorial,thatistheproductofeveryoddnumberfromnto1.3可由定义及卷积公式证明，下同4若X与Y的方差相等，则U与V两者相互独立5TherequirementthatXandYshouldbejointlynormalisessential,withoutitthepropertydoesnothold.Fornon-normalrandomvariablesu

5、ncorrelatednessdoesnotimplyindependence.6当p是奇数时，Ex^p=∫x^p*f(x)dx,奇函数乘以偶函数得奇函数，奇函数在R上的定积分为0Thecentralabsolutemomentscoincidewithplainmomentsforallevenorders,butarenonzeroforoddorders.Foranynon-negativeintegerp,OrderNon-centralmomentCentralmoment1μ02222μ+σσ323μ+3μσ0422444μ+6μσ+3σ3σ53245μ+10μσ

6、+15μσ064224666μ+15μσ+45μσ+15σ15σ7523467μ+21μσ+105μσ+105μσ08624426888μ+28μσ+210μσ+420μσ+105σ105σCombinationoftwoormoreindependentrandomvariablesIfX1andX2aretwoindependentstandardnormalrandomvariableswithmean0andvariance1,thentheirsumanddifferenceisdistributednormallywithmeanzeroandvariancet

7、wo:X1±X2∼N(0,2).IfX1,X2,…,Xnareindependentstandardnormalrandomvariables,thenthesumoftheirsquareshasthechi-squareddistributionwithndegreesoffreedom.IfX1,X2,…,Xnareindependentnormallydistributed2randomvariableswithmeansμandvariancesσ,thentheirsamplemeani