Continuous-Random-Variables-Chapter-5-University-of-South-：連續(xù)型隨機變量5章-中南大學教學課件

ContinuousRandomVariables

Chapter5NutanS.MishraDepartmentofMathematicsandStatisticsUniversityofSouthAlabamaContinuousRandomVariableWhenrandomvariableXtakesvaluesonanintervalForexampleGPAofstudentsX[0,4]HighdaytemperatureinMobileX(20,∞)Recallincaseofdiscretevariablesasimpleeventwasdescribedas(X=k)andthenwecancomputeP(X=k)whichiscalledprobabilitymassfunctionIncaseofcontinuousvariablewemakeachangeinthedefinitionofanevent.ContinuousRandomVariableLetX[0,4],thenthereareinfinitenumberofvalueswhichxmaytake.IfweassignprobabilitytoeachvaluethenP(X=k)0foracontinuousvariableInthiscasewedefineaneventas(x-x

≤X≤x+x)wherexisaverytinyincrementinx.AndthusweassigntheprobabilitytothiseventP(x-x

≤X≤x+x)=f(x)dxf(x)iscalledprobabilitydensityfunction(pdf)Propertiesofpdf(cumulative)DistributionFunctionThecumulativedistributionfunctionofacontinuousrandomvariableisWheref(x)istheprobabilitydensityfunctionofx.

Relationbetweenf(x)andF(x)MeanandVarianceExercise5.2TofindthevalueofkThusf(x)=4x3for0<x<1P(1/4<x<3/4)==P(x>2/3)==Exercise5.7Exercise5.13ProbabilityanddensitycurvesP(a<Y<b):P(100<Y<150)=0.42Usefullink:/faculty/Stefan_Waner/RealWorld/cprob/cprob2.htmlNormalDistributionX=normalrandomvariatewithparametersμandσifitsprobabilitydensityfunctionisgivenby

μandσarecalledparametersofthenormaldistribution/~mjaneba/help/normalcurve.htmlStandardNormalDistributionThedistributionofanormalrandomvariablewithmean0andvariance1iscalledastandardnormaldistribution.StandardNormalDistributionTheletterZistraditionallyusedtorepresentastandardnormalrandomvariable.zisusedtorepresentaparticularvalueofZ.Thestandardnormaldistributionhasbeentabularized.

StandardNormalDistributionGivenastandardnormaldistribution,findtheareaunderthecurve (a)totheleftofz=-1.85 (b)totheleftofz=2.01 (c)totherightofz=–0.99 (d)torightofz=1.50 (e)betweenz=-1.66andz=0.58StandardNormalDistributionGivenastandardnormaldistribution,findthevalueofksuchthat (a)P(Z<k)=.1271 (b)P(Z<k)=.9495 (c)P(Z>k)=.8186 (d)P(Z>k)=.0073 (e)P(0.90<Z<k)=.1806 (f)P(k<Z<1.02)=.1464NormalDistributionAnynormalrandomvariable,X,canbeconvertedtoastandardnormalrandomvariable:

z=(x–μx)/sxUsefullink:

(picturesofnormalcurvesborrowedfrom:/~lynch/509Spring03/25NormalDistributionGivenarandomVariableXhavinganormaldistributionwithμx=10andsx=2,findtheprobabilitythatX<8.4

6810121416zxRelationshipbetweentheNormalandBinomialDistributionsThenormaldistributionisoftenagoodapproximationtoadiscretedistributionwhenthediscretedistributiontakesonasymmetricbellshape.Somedistributionsconvergetothenormalastheirparametersapproachcertainlimits.Theorem6.2:IfXisabinomialrandomvariablewithmeanμ=npandvariances2=npq,thenthelimitingformofthedistributionofZ=(X–np)/(npq).5asn,isthestandardnormaldistribution,n(z;0,1).Exercise5.19UniformdistributionTheuniformdistributionwithparametersαandβhasthedensityfunction

ExponentialDistribution:BasicFactsDensityCDFMeanVarianceKeyProperty:MemorylessnessReliability:AmountoftimeacomponenthasbeeninservicehasnoeffectontheamountoftimeuntilitfailsInter-eventtimes:AmountoftimesincethelasteventcontainsnoinformationabouttheamountoftimeuntilthenexteventServicetimes:Amountofremainingservicetimeisindependentoftheamountofservicetimeelapsedsofar

ExponentialDistribution

Theexponentialdistributionisaverycommonlyuseddistributioninreliabilityengineering.Duetoitssimplicity,ithasbeenwidelyemployedevenincasestowhichitdoesnotapply.Theexponentialdistributionisusedtodescribeunitsthathaveaconstantfailurerate.

Thesingle-parameterexponentialpdfisgivenby:where:

·

λ

=constantfailurerate,infailuresperunitofmeasurement,e.g.failuresperhour,percycle,etc.·

λ

=.1/m·

m=meantimebetweenfailures,ortoafailure.·

T=operatingtime,lifeorage,inhours,cycles,miles,actuations,etc.

Thisdistributionrequirestheestimationofonlyoneparameter,,foritsapplication.JointprobabilitiesFordiscretejointprobabilitydensityfunction(jointpdf)ofak-dimensionaldiscreterandomvariableX=(X1,X2,…,Xk)isdefinedtobef(x1,x2,…,xk)=P(X1=x1,X2=x2,…,Xk=xk)forallpossiblevaluesx=(x1,x2,…,xk)inX.Let(X,Y)havethejointprobabilityfunctionspecifiedinthefollowingtableJointdistributionConsiderJointprobabilitydistributionJointProbabilityDistributionFunction

f(x,y)>0

Marginalpdfofx&y

hereisanexample

x=1,2,3y=1,2Marginalpdfofx&y

Considerthefollowingexample

x=1,2,3

y=1,2IndependentRandomVariablesIfPropertiesofexpectationsforadiscretepdf,f(x),Theexpectedvalueofthefunctionu(x),E[u(X)]=Mean==E[X]=Variance=Var(X)=2=x2=E[(X-)2]=E[X2]-2Foracontinuouspdf,f(x)

E(X)=MeanofX=

E[(X-)2]=E(X2)-[E(X)]2=VarianceofX=PropertiesofexpectationsE(aX+b)=aE(X)+bVa