第3章概率分布

第3章概率分布试验得到的数据通常呈现一定的规律性，引入随机变量以后，可以将随机数据表达为随机变量的函数。常见的随机变量有离散性随机变量和连续性随机变量两种。当变量全部可以取得的值是有限个或可列无限多个的时候，称为离散性随机变量；如果对于随机变量X的分布函数()Fx，存在非负函数()fx，使得对于任何实数x有：()()xFxftdt（1）则称x为连续型随机变量。对应于离散型随机变量和连续型随机变量，有离散型概率分布函数和连续型分布函数。第一节概率密度函数一、基本数学原理对于离散型概率分布和连续型概率分布，二者的概率密度函数定义有所不同。（1）中的函数()fx称为X的概率密度函数。该函数具有以下性质：（1）()0fx（2）()1fxdx；（3）211221{}()()()xxPxXxFxFxfxdx；（4）若()fx在点x处连续，则有'()()Fxfx。对于离散型概率分布，则不称其为概率密度函数，而称为概率分布或者分布律。设离散型随机变量X所有可能取的值为(1,2,...)kxk，x取各个不同可能值的概率，即事件{}kXx的概率为：{}kkPXxp，1,2,...k。kp即称为分布律。具有下面性质：（1）0kp，1,2,...k；（2）11kkp；二、相关函数介绍概率密度函数的名称采用PDF.distriname的形式，这里PDF表示ProbabilityDensityFunction,distriname是分布的名称，比如正态分布的概率密度函数的名称用函数PDF.NORMAL()计算不同分布的概率分布密度计算函数如下表1所示：表1：ProbabilityDensityFunctionsThefollowingfunctionsgivethevalueofthedensityfunctionwiththespecifieddistributionatthevaluequant,thefirstargument.Subsequentargumentsaretheparametersofthedistribution.Notetheperiodineachfunctionname.PDF.BERNOULLI.PDF.BERNOULLI(quant,prob)Numeric.ReturnstheprobabilitythatavaluefromtheBernoullidistribution,withthegivenprobabilityparameter,willbeequaltoquant.PDF.BETA.PDF.BETA(quant,shape1,shape2)Numeric.Returnstheprobabilitydensityofthebetadistribution,withthegivenshapeparameters,atquant.PDF.BINOM.PDF.BINOM(quant,n,prob)Numeric.Returnstheprobabilitythatthenumberofsuccessesinntrials,withprobabilityprobofsuccessineach,willbeequaltoquant.Whennis1,thisisthesameasPDF.BERNOULLI.PDF.BVNOR.PDF.BVNOR(quant1,quant2,corr)Numeric.Returnstheprobabilitydensityofthestandardbivariatenormaldistribution,withthegivencorrelationparameter,atquant1,quant2.PDF.CAUCHY.PDF.CAUCHY(quant,loc,scale)Numeric.ReturnstheprobabilitydensityoftheCauchydistribution,withthegivenlocationandscaleparameters,atquant.PDF.CHISQ.PDF.CHISQ(quant,df)Numeric.Returnstheprobabilitydensityofthechi-squaredistribution,withdfdegreesoffreedom,atquant.PDF.EXP.PDF.EXP(quant,shape)Numeric.Returnstheprobabilitydensityoftheexponentialdistribution,withthegivenshapeparameter,atquant.PDF.F.PDF.F(quant,df1,df2)Numeric.ReturnstheprobabilitydensityoftheFdistribution,withdegreesoffreedomdf1anddf2,atquant.PDF.GAMMA.PDF.GAMMA(quant,shape,scale)Numeric.Returnstheprobabilitydensityofthegammadistribution,withthegivenshapeandscaleparameters,atquant.PDF.GEOM.PDF.GEOM(quant,prob)Numeric.Returnstheprobabilitythatthenumberoftrialstoobtainasuccess,whentheprobabilityofsuccessisgivenbyprob,willbeequaltoquant.PDF.HALFNRM.PDF.HALFNRM(quant,mean,stddev)Numeric.Returnstheprobabilitydensityofthehalfnormaldistribution,withspecifiedmeanandstandarddeviation,atquant.PDF.HYPER.PDF.HYPER(quant,total,sample,hits)Numeric.Returnstheprobabilitythatthenumberofobjectswithaspecifiedcharacteristic,whensampleobjectsarerandomlyselectedfromauniverseofsizetotalinwhichhitshavethespecifiedcharacteristic,willbeequaltoquant.PDF.IGAUSS.PDF.IGAUSS(quant,loc,scale)Numeric.ReturnstheprobabilitydensityoftheinverseGaussiandistribution,withthegivenlocationandscaleparameters,atquant.PDF.LAPLACE.PDF.LAPLACE(quant,mean,scale)Numeric.ReturnstheprobabilitydensityoftheLaplacedistribution,withthespecifiedmeanandscaleparameters,atquant.PDF.LOGISTIC.PDF.LOGISTIC(quant,mean,scale)Numeric.Returnstheprobabilitydensityofthelogisticdistribution,withthespecifiedmeanandscaleparameters,atquant.PDF.LNORMAL.PDF.LNORMAL(quant,a,b)Numeric.Returnstheprobabilitydensityofthelog-normaldistribution,withthespecifiedparameters,atquant.PDF.NEGBIN.PDF.NEGBIN(quant,thresh,prob)Numeric.Returnstheprobabilitythatthenumberoftrialstoobtainasuccess,whenthethresholdparameteristhreshandtheprobabilityofsuccessisgivenbyprob,willbeequaltoquant.PDF.NORMAL.PDF.NORMAL(quant,mean,stddev)Numeric.Returnstheprobabilitydensityofthenormaldistribution,withspecifiedmeanandstandarddeviation,atquant.PDF.PARETO.PDF.PARETO(quant,threshold,shape)Numeric.ReturnstheprobabilitydensityoftheParetodistribution,withthespecifiedthresholdandshapeparameters,atquant.PDF.POISSON.PDF.POISSON(quant,mean)Numeric.ReturnstheprobabilitythatavaluefromthePoissondistribution,withthespecifiedmeanorrateparameter,willbeequaltoquant.PDF.T.PDF.T(quant,df)Numeric.ReturnstheprobabilitydensityofStudent'stdistribution,withthespecifieddegreesoffreedomdf,atquant.PDF.UNIFORM.PDF.UNIFORM(quant,min,max)Numeric.Returnstheprobabilitydensityoftheuniformdistribution,withthespecifiedminimumandmaximum,atquant.PDF.WEIBULL.PDF.WEIBULL(quant,a,b)Numeric.ReturnstheprobabilitydensityoftheWeibulldistribution,withthespecifiedparameters,atquant.NPDF.BETA.NPDF.BETA(quant,shape1,shape2,nc)Numeric.Returnstheprobabilitydensityofthenoncentralbetadistribution,withthegivenshapeandnoncentralityparameters,atquant.NPDF.CHISQ.NPDF.CHISQ(quant,df,nc)Numeric.Returnstheprobabilitydensityofthenoncentralchi-squaredistribution,withdfdegreesoffreedomandthespecifiednoncentralityparameter,atquant.NPDF.F.NPDF.F(quant,df1,df2,nc)Numeric.ReturnstheprobabilitydensityofthenoncentralFdistribution,withdegreesoffreedomdf1anddf2andnoncentralitync,atquant.NPDF.T.NPDF.T(quant,df,nc)Numeric.ReturnstheprobabilitydensityofthenoncentralStudent