概率论与数理统计(英文)第四章.doc

4.ContinuousRandomVariable连续型随机变量

Continuousrandomvariablesappearwhenwedealwillquantitiesthataremeasuredonacontinuousscale.Forinstance,whenwemeasurethespeedofacar,theamountofalcoholinapersonsblood,thetensilestrengthofnewalloy.

Weshalllearnhowtodetermineandworkwithprobabilitiesrelatingtocontinuousrandomvariablesinthischapter.Weshallintroducetotheconceptoftheprobabilitydensityfunction.

4.1ContinuousRandomVariable

1.Definition

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

(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

SolutionPerfo