概率论与数理统计英文版总结Word下载.doc

概率论与数理统计英文版总结Word下载.doc

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Theorem2.4.1Ifandaremutuallyexclusive,then

（2.4.1）

Mutuallyindependent事件的独立性

Twoeventsandaresaidtobeindependentif

OrTwoeventsandareindependentifandonlyif

.

ConditionalProbability条件概率

Theprobabilityofaneventisfrequentlyinfluencedbyotherevents.

DefinitionTheconditionalprobabilityof,given,denotedby,isdefinedby

if.（2.5.1）

Themultiplicationtheorem乘法定理

Ifareevents,then

Iftheeventsareindependent,thenforanysubset,

（全概率公式totalprobability）

Theorem2.6.1.IftheeventsconstituteapartitionofthesamplespaceSsuchthatforthanforanyeventof,

（2.6.2）

（贝叶斯公式Bayes’formula.）

Theorem2.6.2IftheeventsconstituteapartitionofthesamplespaceSsuchthatforthanforanyeventAofS,,

.for（2.6.2）

ProofBythedefinitionofconditionalprobability,

Usingthetheoremoftotalprobability,wehave

1.randomvariabledefinition

Definition3.1.1Arandomvariableisarealvaluedfunctiondefinedonasamplespace;

i.e.itassignsarealnumbertoeachsamplepointinthesamplespace.

2.Distributionfunction

Definition3.1.2Letbearandomvariableonthesamplespace.Thenthefunction

.

iscalledthedistributionfunctionof

NoteThedistributionfunctionisdefinedonrealnumbers,notonsamplespace.

3.Properties

Thedistributionfunctionofarandomvariablehasthefollowingproperties:

（1）isnon-decreasing.

Infact,if,thentheeventisasubsetoftheevent,thus

（2）,

.

（3）Forany,.Thisistosay,thedistributionfunctionofarandomvariableisrightcontinuous.

3.2DiscreteRandomVariables离散型随机变量

Definition3.2.1Arandomvariableiscalledadiscreterandomvariable,ifittakesvaluesfromafinitesetor,asetwhoseelementscanbewrittenasasequence

geometricdistribution（几何分布）

X

1

2

3

4

…

k

P

p

q1p

q2p

q3p

qk－1p

Binomialdistribution（二项分布）

Definition3.4.1ThenumberofsuccessesinBernoullitrialsiscalledabinomialrandomvariable.Theprobabilitydistributionofthisdiscreterandomvariableiscalledthebinomialdistributionwithparametersand,denotedby.

poissondistribution（泊松分布）

Definition3.5.1AdiscreterandomvariableiscalledaPoissonrandomvariable,ifittakesvaluesfromtheset,andif

（3.5.1）

Distribution（3.5.1）iscalledthePoissondistributionwithparameter,denotedby.

Expectation（mean）数学期望

Definition3.3.1Letbeadiscreterandomvariable.Theexpectationormeanofisdefinedas

（3.3.1）

2．Variance方差standarddeviation（标准差）

Definition3.3.2Letbeadiscreterandomvariable,havingexpectation.Thenthevarianceof,denotebyisdefinedastheexpectationoftherandomvariable

（3.3.6）

Thesquarerootofthevariance,denoteby,iscalledthestandarddeviationof:

（3.3.7）

probabilitydensityfunction概率密度函数

Definition4.1.1Afunctionf（x）definedoniscalledaprobabilitydensityfunction（概率密度函数）if:

（i）;

（ii）f（x）isintergrable（可积的）onand.

Definition4.1.2

Letf（x）beaprobabilitydensityfunction.IfXisarandomvariablehavingdistributionfunction

（4.1.1）

thenXiscalledacontinuousrandomvariablehavingdensityfunctionf（x）.Inthiscase,

.（4.1.2）

5.Mean（均值）

Definition4.1.2LetXbeacontinuousrandomvariablehavingprobabilitydensityfunctionf（x）.Thenthemean（orexpectation）ofXisdefinedby

（4.1.3）

providedtheintegralconvergesabsolutely.

6.variance 方差

Similarly,thevarianceandstandarddeviationofacontinuousrandomvariableXisdefinedby

（4.1.4）

WhereisthemeanofX,isreferredtoasthestandarddeviation.

Weeasilyget

.（4.1.5）

.

4.2UniformDistribution均匀分布

Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunction

4.5ExponentialDistribution指数分布Definition4.5.1AcontinuousvariableXhasanexponentialdistributionwithparameter,ifitsdensityfunctionisgivenby

（4.5.1）

Theorem4.5.1ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenby

4.3NormalDistribution正态分布

1.Definition

Theequationofthenormalprobabilitydensity,whosegraphisshowninFigure4.3.1,is

4.4NormalApproximationtotheBinomialDistribution（二项分布）

nislarge（n>

30）,piscloseto0.50,

4.7Chebyshev’sTheorem（切比雪夫定理）

Theorem4.7.1Ifaprobabilitydistributionhasmeanμandstandarddeviationσ,theprobabilityofgettingavaluewhichdeviatesfromμbyatleastkσisatmost.Symbolically,

Jointprobabilitydistribution（联合分布）

Inthestudyofprobability,givenatleasttworandomvariablesX,Y,...,thataredefinedonaprobabilityspace,thejointprobabilitydistributionforX,Y,...isaprobabilitydistributionthatgivestheprobabilitythateachofX,Y,...fallsinanyparticularrangeordiscretesetofvaluesspecifiedforthatvariable.

5.2Conditionaldistribution条件分布

ConsistentwiththedefinitionofconditionalprobabilityofeventswhenAistheeventX=xandBistheeventY=y,theconditionalprobabilitydistributionofXgivenY=yisdefinedas

forallxprovided.

5.3Statisticalindependent随机变量的独立性

Definition5.3.1Supposethepair{X,Y}ofrealrandomvariableshasjointdistributionfunctionF（x,y）.IftheF（x,y）obeytheproductrule

forallx,y.

thetworandomvariablesXandYareindependent,orthepair{X,Y}isindependent.

5.4CovarianceandCorrelation协方差和相关系数

WenowdefinetworelatedquantitieswhoseroleincharacterizingtheinterdependenceofXandYwewanttoexamine.

Definition5.4.1SupposeXandYarerandomvariables.Thecovarianceofthepair{X,Y}is

Thecorrelationcoefficientofthepair{X,Y}is

Where

Definition5.4.2TherandomvariablesXandYaresaidtobeuncorrelatediff.

5.5LawofLargeNumbersandCentralLimitTheorem中心极限定理

Wecanfindthesteadilyofthefrequencyoftheeventsinlargenumberofrandomphenomenon.Andtheaverageoflargenumberofrandomvariablesarealsosteadiness.Theseresultsarethelawoflargenumbers.

Theorem5.5.1Ifasequenceofrandomvariablesisindependent,with

then

.（5.5.1）

Theorem5.5.2LetequalsthenumberoftheeventAinnBernoullitrials,andpistheprobabilityoftheeventAonanyoneBernoullitrial,then

.（5.5.2）

（频率具有稳定性）

Theorem5.5.3Ifisindependent,with

then.

population（总体）

Definition6.2.1Apopulationisthesetofdataormeasurementsconsistsofallconceivablypossibleobservationsfromallobjectsinagivenphenomenon.

Apopulationmayconsistoffinitelyorinfinitelymanyvarieties.

sample（样本、子样）

Definition6.2.2Asampleisasubsetofthepopulationfromwhichpeoplecandrawconclusionsaboutthewhole.

sampling（抽样）

takingasample:

Theprocessofperforminganexperimenttoobtainasamplefromthepopulationiscalledsampling.

中位数

Definition6.2.4Ifarandomsamplehastheorderstatistics,then

（i）TheSampleMedianis

（ii）TheSampleRangeis

SampleDistributions抽样分布

1．samplingdistributionofthemean均值的抽样分布

Theorem6.3.1Ifismeanoftherandomsampleofsizefromarandomvariablewhichhasmeanandthevariance,then

and.

Itiscustomarytowriteasandas.

Here,iscalledtheexpectationofthemean.均值的期望

iscalledthestandarderrorofthemean.均值的标准差

7.1PointEstimate点估计

Definition7.1.1Supposeisaparameterofapopulation,isarandomsamplefromthispopulation,andisastatisticthatisafunctionof.Now,totheobservedvalue,ifweuseasanestimatedvalueof,theniscalledapointestimatorofandisreferredasapointestimateof.Thepointestimatorisalsooftenwrittenas.

Unbiasedestimator（无偏估计量）

Definition7.1.2.Supposeisanestimatorofaparameter.Thenisunbiasedifandonlyif

minimumvarianceunbiasedestimator（最小方差无偏估计量）

Definition7.1.3Letbeanunbiasedestimatorof.Ifforanywhichisalsoanunbiasedestimatorof,wehave

theniscalledtheminimumvarianceunbiasedestimatorof.Sometimesitisalsocalledbestunbiasedestimator.

3.MethodofMoments矩估计的方法

Definition7.1.4

SupposeconstitutearandomsamplefromthepopulationXthat