概率论与数理统计英文第四部分.docx

概率论与数理统计英文第四部分.docx

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概率论与数理统计英文第四部分

4.ContinuousRandomVariable连续型随机变量

Continuousrandomvariablesappearwhenwedealwillquantitiesthataremeasuredonacontinuousscale.Forinstance,whenwemeasurethespeedofacar,theamountofalcoholinaperson'sblood,thetensilestrengthofnewalloy.

Weshalllearnhowtodetermineandworkwithprobabilitiesrelatingtocontinuousrandomvariablesinthischapter.Weshallintroducetotheconceptoftheprobabilitydensityfunction.

4.1ContinuousRandomVariable

1.Definition

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）

2.几何意义

3.Note

Inmostapplications,f（x）iseithercontinuousorpiecewisecontinuoushavingatmostfinitelymanydiscontinuities.

Note1ForarandomvariableX,wehaveadistributionfunction.IfXisdiscrete,ithasaprobabilitydistribution.IfXiscontinuous,ithasaprobabilitydensityfunction.

Note2LetXbeacontinuousrandomvariable,thenforanyrealnumberx,

.

4.Example

Example4.1.2

Findksothatthefollowingcanserveastheprobabilitydensityofacontinuousrandomvariable:

SolutionTosatisfytheconditions（4.1.1）,kmustbenonnegative,andtosatisfythecondition（4.1.2）wemusthave

sothat.

（Cauchydistribution柯西分布）

□

Example4.1.3Calculatingprobabilitiesfromtheprobabilitydensityfunction

Ifarandomvariablehastheprobabilitydensity

Findtheprobabilityfromthatitwilltakeonvalue

（a）between0and2;

（b）greaterthan1.

SolutionEvaluatingthenecessaryintegrals,weget

（a）

（b）

Example4.1.4

DeterminingthedistributionfunctionofX,itisknown

SolutionPerformingthenecessaryintegrations,weget

□

5.mean

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

（4.1.3）

providedtheintegralconvergesabsolutely.

Iftheintegral（4.1.3）doesnotconvergesabsolutely（绝对收敛）,wesaythemeanofXdoesnotexist.

Themeanofcontinuousrandomvariablehasthesimilarpropertiesasdiscreterandomvariable.

Ifg（X）isanintegrablefunctionofacontinuousrandomvariableX,havingdensityfunctionf（x）,meanofg（X）is

providedtheintegralconvergesabsolutely.

Example4.6.4

LetXbearandomvariablehavingCauchydistribution,theprobabilitydensityfunctionisgivenby

（a）FindE（X）;

（b）Let

Find.

Solution（a）Sincetheintegraldiverges（发散）,E（X）doesnotexist.

（b）

6.variance

Similarly,thevarianceandstandarddeviationofacontinuousrandomvariableXisdefinedby

（4.1.4）

WhereisthemeanofX,isreferredtoasthestandarddeviation.

Weeasilyget

.（4.1.5）

Example4.1.5

Determiningthemeanandvarianceusingtheprobabilitydensityfunction

Withreferencetotheexample4.1.3:

findthemeanandvarianceofthegivenprobabilitydensity.

SolutionPerformingthenecessaryintegrations,weget

and

.

4.2UniformDistribution均匀分布

Theuniformdistribution,withtheparametersaandb,hasprobabilitydensityfunction

whosegraphisshowninFigure4.2.1.

Figure4.2.1Theuniformprobabilitydensityintheinterval（a,b）

Toproof.

Tofindthedistributionfunction.

Thedistributionfunctionoftheuniformdistributionis

Notethatallvaluesofxfromaandbare“equallylikely”,inthesensethattheprobabilitythatxliesinanintervalofwidthentirelycontainedintheintervalfromatobisequalto,regardlessoftheexactlocationoftheinterval.

Tofindthemeanandvarianceoftheuniformdistribution

And

EX2=

Thus,

4.5ExponentialDistribution指数分布

Definition4.5.1AcontinuousvariableXhasanexponentialdistributionwithparameter,ifitsdensityfunctionisgivenby

（4.5.1）

Manyrandomvariables,suchasthelifeofautomotiveparts,lifeofanimals,timeperiodbetweentwocallsarrivestoanoffice,havingadistributioncalledexponentialdistribution.

NoteEquation（4.5.1）reallygivesadensityfunction,since

Theorem4.5.1ThemeanandvarianceofacontinuousrandomvariableXhavingexponentialdistributionwithparameterisgivenby

.

ProofSincetheprobabilitydensityfunctionofXis（4.5.1）,wehave

Example4.5.1.

AssumethatthelifeYofbulbsproducedbycompanyXhasexponentialdistributionwithmean.

（a）FindtheprobabilitythatabulbselectedatrandomfromtheproductofcompanyXhaslifelongerthan450hrs.

（b）Select5bulbsrandomlyfromtheproductofcompanyX,whatistheprobabilitythatatleast3ofthemhaslifelongerthan450hrs.

S