钱小军《概率论与数理统计》chap05-binomialdistribu

2020/4/30,Tsinghua-CUHKFinanceMBA2008,1,第五章离散型概率分布Chapter5DiscreteProbabilityDistribution,钱小军清华大学经济管理学院伟伦楼南456B电话:62789934电子邮箱：qianxj,2020/4/30,Tsinghua-CUHKFinanceMBA2008,2,1.RandomVariables随机变量,RandomExperiment随机试验Arandomvariableisanumericaldescriptionoftheoutcomeofanexperiment.随机变量是一次随机试验结果的数值性描述。Arandomvariableisavariablewhosevaluevarieswithdifferentoutcomesofrandomexperimentandisofdifferentprobabilities.随机变量是一个变量，它的取值随随机试验的结果不同而不同，并以一定的概率取这些不同的数值。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,3,ArandomvariablecanbeclassifiedasbeingeitherDiscreteRandomVariableorContinuousRandomVariableonthenumericalvaluesitassumes.随机变量根据所代表的数值，可定义为离散型随机变量或连续型随机变量。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,4,DiscreteRandomVariable离散型随机变量,Arandomvariablethatmayassumeeitherafinitenumberofvaluesoraninfinitesequenceofvaluessuchas0,1,2,isreferredtoasadiscreterandomvariable.离散型随机变量是只取有限个不同的数值或无限个但可按顺序列出的数值如0,1,2,的随机变量。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,5,举例Example（中文P120，练习1EnglishP178Exercises1）,Considertheexperimentoftossingacointwice,andwedefinearandomvariablethatrepresentsthenumberofheadsoccurringonthetwicetosses.抛两次硬币，我们定义一个随机变量表示出现正面的次数,或,2020/4/30,Tsinghua-CUHKFinanceMBA2008,6,2.DiscreteProbabilityDistribution离散型概率分布,Theprobabilitydistributionforarandomvariabledescribeshowprobabilitiesaredistributedoverthevaluesoftherandomvariable.随机变量的概率分布描述随机变量所取的所有不同的值以及取这些数值的概率。Foradiscreterandomvariable,theprobabilitydistributionisdefinedbyaprobabilityfunction,denotedby.对于离散型随机变量，概率分布通过概率函数来定义。Theprobabilityfunctionprovidestheprobabilityforeachvalueoftherandomvariable.概率函数提供了随机变量取每一个值的概率。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,7,Theessentialconditionofdiscreteprobabilityfunction离散型概率函数的必要条件：1.2.,2020/4/30,Tsinghua-CUHKFinanceMBA2008,8,.,Example举例1：抛两次硬币的试验,2020/4/30,Tsinghua-CUHKFinanceMBA2008,9,Example举例2：(中文版:P121，表5-3）(EnglishVersion:P180Table5-3),DiCarlo汽车公司汽车日销售量的概率分布,2020/4/30,Tsinghua-CUHKFinanceMBA2008,10,GraphicPresentationofDiscreteProbabilityFunction离散型概率函数的图形表示,连续抛两次硬币出现正面的次数,0.25,0.50,0,1,2,x,f(x),2020/4/30,Tsinghua-CUHKFinanceMBA2008,11,MeasureoftheDiscreteRandomVariable离散型随机变量的度量,Expectation(orMean)数学期望（或均值）Variance方差StandardDeviation标准差,2020/4/30,Tsinghua-CUHKFinanceMBA2008,12,Expectation(orMean)数学期望,Theexpectedvalueofarandomvariableisameasureofthecentrallocationfortherandomvariable随机变量的数学期望是对随机变量取值的中心位置的一种度量。Formula公式：,2020/4/30,Tsinghua-CUHKFinanceMBA2008,13,Example举例：P124，DiCarlo汽车公司汽车日销售量数学期望的计算(期望.xls),2020/4/30,Tsinghua-CUHKFinanceMBA2008,14,(1)(2)(3),PropertiesofExpectation数学期望的性质,E是可以“分配”的？保持线性关系不变。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,15,Varianceisameasureofvariability,ordispersionforarandomvariable.方差用来描述随机变量的变异性或分散度。VarianceofaDiscreteRandomVariable离散型随机变量的方差：,2020/4/30,Tsinghua-CUHKFinanceMBA2008,16,Example举例：DiCarlo汽车公司汽车日销售量的方差的计算(方差的计算.xls),2020/4/30,Tsinghua-CUHKFinanceMBA2008,17,Example举例：DiCarlo汽车公司汽车日销售量的方差的计算(方差的计算.xls),2020/4/30,Tsinghua-CUHKFinanceMBA2008,18,(1)(2)(3),PropertiesofVariance方差的性质,如果都是独立（不相关）的，,2020/4/30,Tsinghua-CUHKFinanceMBA2008,19,StandardDeviation标准差,StandardDeviationisdefinedasthepositivesquarerootoftheVariance.标准差定义为方差的非负平方根。Standarddeviationhasthesameunitastherandomvariable,soitalwaysdescribesthevariabilityforarandomvariable.标准差的单位与随机变量的相同，因而更常用来描述一个随机变量的变异性（变化幅度）。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,20,4.BinomialProbabilityDistribution二项概率分布,BinomialExperiment二项试验Theexperimentconsistsofasequenceofnidenticaltrials.试验由一个包括n次相同试验的序列组成;Thetrialsareindependent.每次试验是独立的。Twooutcomesarepossibleoneachtrial.Werefertooneasasuccessandtheotherasthefailure.每次试验只有两种可能结果：成功或失败;Theprobabilityofasuccess,denotedbyp,doesnotchangefromtrailtotrail.Consequently,theprobabilityofafailure,denotedby1-p,doesnotchangefromtrailtotrial.成功的概率，用p表示，失败的概率用1-p表示;,2020/4/30,Tsinghua-CUHKFinanceMBA2008,21,考虑三个进入马丁服装店的顾客的购买决定。根据经验，任一顾客购买服装的概率为0.3。三个顾客中有两个会购买服装的概率是多少？,Example举例：马丁服装店问题(中文版:P128)(EnglishVersion:P190）,2020/4/30,Tsinghua-CUHKFinanceMBA2008,22,Checkingthefourrequirementsforabinomialexperiment检查是否满足二项试验的属性,Theexperimentcanbedescribedasasequenceofthreeidenticaltrials,onetrialforeachofthethreecustomerswhowillenterthestore.试验可以被认为由一个包括三个相同试验的序列构成，任一顾客进入商店为一次试验Twooutcomes-thecustomermakesaspurchase(success)orthecustomerdoesnotmakeapurchase(failure)-arepossibleforeachtrial.每次试验都有两个结果：顾客购买(成功)或不购买(失败),2020/4/30,Tsinghua-CUHKFinanceMBA2008,23,Theprobabilitythatcustomerwillmakeapurchase(0.30)orwillnotmakeapurchase(0.70)isassumedtobethesameforallcustomers.顾客购买的概率(0.3)或不购买的概率(0.7)对所有顾客都是一样的Thepurchasedecisionofeachcustomerisindependentofthedecisionsoftheothercustomers.每个顾客的购买决定不受其他顾客的购买决定影响（独立的）Thevariableisbinomiallydistributedwithn=3andp=0.3.这个随机变量是n=3,p=0.3的二项分布。,Continued(续),2020/4/30,Tsinghua-CUHKFinanceMBA2008,24,NumberofExperimentalOutcomesProvidingExactlyxSuccessesinnTrialsn次试验中恰有x次成功的试验数目：BinomialProbabilityFunction二项概率函数n次试验中有x次成功的概率，其中p是一次试验中成功的概率。pistheprobabilityofsuccessinonetrial.,2020/4/30,Tsinghua-CUHKFinanceMBA2008,25,Probabilitydistributionforthenumberofcustomersmakingapurchase马丁服装店：进行购买的顾客人数的概率分布,2020/4/30,Tsinghua-CUHKFinanceMBA2008,26,二项概率表.xls中文书参加附赠CD里面的统计表格英文书A12,UsingTablesofBinomialProbabilities二项概率表的使用,2020/4/30,Tsinghua-CUHKFinanceMBA2008,27,2020/4/30,Tsinghua-CUHKFinanceMBA2008,28,0-1distribution0-1分布,n=1时的二项分布称为01分布，即表示成功。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,29,0-1distributionandbinomialdistribution0-1分布与二项分布,如果是服从成功的概率为p的0-1分布，而且相互独立，则即服从二项分布。反过来，如果，则可以认为是n个独立的0-1分布的和，其中成功的概率为p。,2020/4/30,Tsinghua-CUHKFinanceMBA2008,30,ExpectationandVariancefortheBinomialProbabilityDistribution二项分布的数学期望和方差,如果随机变量，则,2020/4/30,Tsinghua-CUHKFinanceMBA2008,31,Example举例：马丁服装店,3个顾客：1000个顾客：问题：怎样增加销售量的期望值或平均销售量？,2020/4/30,Tsinghua-CUHKFinanc