概率论与数理统计：chapter3 discrete variable

3DiscreteRandomVariablesandProbabilityDistributions3.1RandomVariablesForagivensamplespaceSofsomeexperiment,arandomvariableisanyrulethatassociatesanumberwitheachoutcomeinS.DefinitionUserv

inplaceofrandomvariable.Randomvariablesaredenotedbyuppercaseletters,suchasXandY,xtorepresentsomeparticularvalueofthecorrespondingrandomvariable.Definition

Anyrandomvariablewhoseonlypossiblevaluesare0and1iscalledaBernoullirandomvariable.

wecandefineX=1whenatailisobservedandX=0whenaheadisobserved.Thedefinitionisarbitrarybutitmustbefixedbeforetheexperimentisstarted.WewilloftenwanttodefineandstudyseveraldifferentrandomvariablesfromthesamesamplespaceExample1Thereare5ballsinthebag,inwhich2whiteballsand3blackballs.Selectthreeballsonetimeatrandommanner.Solution:SupposeX={numberofselectedwhiteballs}Example2Rolltwodice,letXdenotedthesumoftwodiceshownumber.Solution:Example3Considertheexperimentoftossingacoinfivetimesandoneachtossobservingwhetherthecoinlandswithaheadortailonitsupwardface.TwoTypeOfRandomVariablesDefinition

Adiscreterandomvariableisanrvwhosepossiblevalueseitherconstituteafinitesetorelsecanbelistedinaninfinitesequenceinwhichthereisafirstelement,asecondelement,andsoon.Arandomvariableiscontinuousifitssetofpossiblevaluesconsistsofanentireintervalonthenumberline.

3.2ProbabilityDistributionsForDiscreteRandomVariablesDefinitionTheprobabilitydistributionorprobabilitymassfunction(Pmf)ofadiscretervisdefinedforeverynumberxbyP(x)=P(X=x)

WhenprobabilitiesareassignedtovariousoutcomesinS,theseinturndetermineprobabilitiesassociatedwiththevaluesofanyparticularrvX.theprobabilitydistributionofXsayshowthetotalprobabilityof1isdistributedamong(allocatedto)thevariouspossibleXvalues.Theconditions1P(x)0

2=1Xx1

x2

…pp1

p2

…Example1Thereare5ballsinthebag,inwhich2whiteballsand3blackballs.Selectthreeballsonetimeatrandommanner,determinetheprobabilitydistributefunctionofthenumberofselectedballsbeingwhiteSolution:supposex={numberofselectedwhiteballs}TheprobabilitydistributionfunctionX012p1/106/103/10Example2:thefrequencyfunction

isX-10123p0.16a/10a22a/100.3Determinetheparametera?Solution:duetopropertyofthefrequencyfunction

0.16+a/10+a2+2a/10+0.3=1(a+0.9)(a-0.6)=0a=0.6Example3Rolltwodice,letXdenotedthesumoftwodiceshownumber.DeterminetheprobabilitydistributionofX?Solution:

X23456789101112p1/362/363/364/365/366/365/364/363/362/361/36Example4Someoneusenkeystoopenadoor,butonlyonekeycanopenthedoor,sohetryonebyone.LetXdenotedtrytimestillthedoorbeopened.PleasewritetheprobabilitydistributionofXunderfollowcondition.a.Thekeysthatcouldnotopenthedoorwillputaway.b.Thekeysthatcouldnotopenthedoorwillnotbeputaway.Solution:

a.X123……..np1/n1/n1/n……..1/nb.Example5Thedefectiverateofautomaticproductlineisp.Whenadefectiveproductswasproduced,theautomaticproductlinemustbeadjusted.LetXdenotedthenumberofgoodproductsduringtwotimesadjustment.PleasedeterminetheprobabilitydistributionofX.Solution:Example6Considertheexperimentoftossingacoinfivetimesandoneachtossobservingwhetherthecoinlandswithaheadortailonitsupwardface.Discussthisexperimentwithfourpropertiessatisfied.AParameterOfAProbabilityDistributionDefinition

SupposeP(x)dependsonaquantitythatcanbeassignedanyoneofanumberofpossiblevalues,witheachdifferentvaluedeterminingadifferentprobabilitydistribution.suchaquantityiscalledaaparameterofthedistribution.thecollectionofallprobabilitydistributionsfordifferentvaluesoftheaparameteriscalledafamilyofaprobabilitydistributions.TheCumulativeDistributionFunctionDefinitionThecumulativedistributionfunction(cdf)F(x)ofadiscretervXwithpmfP(x)isdefinedforeverynumberXbyF(x)=P(Xx)=Foranynumberx,F(x)istheprobabilitythattheobservedvalueofXwillbeatmostx.ForXadiscreterv,thegraphofF(x)willhaveajumpateverypossiblevalueofXandwillbeflatbetweenpossiblevalues.SuchagraphiscalledastepfunctionExample1Example2Property1thecdfisnon-decreasing

Property3satisfies:Property2Definition

Foranytwonumbersaandbwithab,P(aXb)=F(b)-F(a-)AnotherViewOfProbabilityMassFunction3.3ExpectedValuesOfDiscreteRandomVariablesTheExpectedValuesOfXDefinition

LetXbeadiscretervwithsetofpossiblevaluesDandpmf

P(x).theexpectedvaluesormeanvalueofX,denotedbyE(X)or,isE(X)==Providedthatifthesumdiverges,theexpectationisundefined.Example

DeterminetheexpectednumberofspotsYthatshowwhenafairdieisrolledinanhonestmanner.Example

ExpectationofaGeometricRandomVariable

X~G(p),averageExample:Onepeopleusenkeystoopenaparticulardoor,andonlyonekeycanopenthedoor,buthedoesnotknowwhichone,hetriesonebyone,tillthedooropened.LetXdenotethenumberofhistries.PleasecalculateEXontheconditionof(1)thekeysunopenedthedoorareputaway;(2)thekeysunopenedthedoorarenotputaway.Solution:(1)theprobabilityfunctionisP(X=k)=1/n,k=1,2,…,nSo,(2)theprobabilityfunctionisExample:Inashootinggame,eachshootermayshootfourtimes.Supposeifnobullethitthetarget,thentheshooterscoreiszero;ifonebullethit,thescoreis15;iftwobulletshit,thescoreis30;ifthreebulletshit,thescoreis55;andifallbulletshit,thescoreis100.Theprobabilityoftheshooterhittingthetargetis0.6,calculatehisexpectedscore.X0153055100PEX=44.52Solution:letXdenotehisscore,thentheprobabilityfunctionis

TheExpectedValuesOfAFunction

DefinitionletXbeadiscretervwithsetofpossiblevaluesDandpmf

p(x).theexpectedvaluesormeanvalueofanyfunctionh(X),denotedbyE[h(X)]or,isEh[(X)]==Xx1x2……xnPp1p2……pnh(X)h(x1)h(x2)……h(xn)Pp1p2……pnRulesOfExpectedValue

Proposition

E(aX+b)=aE(X)+bTheVarianceOfXDefinition

letXhavepmfP(x)andexpectedvalue.ThenthevarianceofX,denotedbyV(X)or,orjust,isV(X)==Thestandarddeviation(SD)ofXisAShortcutFormulaForPropositionV(X)===－－RulesOfVarianceProposition

V(aX+b)=3.4theBinomialprobabilityDistributionPropertiesofaBinomialexperiment

1.Theexperimentconsistsofasequenceofnidenticaltrials.2.Twooutcomesarepossibleoneachtrial.Werefertooneasasuccess(S)andtheotherasafailure(F).3.Theprobabilityofasuccess,denotedby

p,doesnotchangefromtrialtotrial.Consequently,theprobabilityofafailure,denotedby1-p,doesnotchangefromtrialtotrial.4.Thetrialsareindependent.(Trialsareindependentfromonetoanother)Example1Considertheexperimentoftossingacoinfivetimesandoneachtossobservingwhetherthecoinlandswithaheadortailonitsupwardface.Discussthisexperimentwithfourpropertiessatisfied.Definition:Anexperimentforwhichconditions1-4aresatisfiediscalledabinomialexperiment.Manyexperimentsinvolveasequenceofindependenttrialsforwhichtherearemorethantwopossibleoutcomesonanyonetrial.Abinomialexperimentcanbecreatedbydividingthepossibleoutcomesintotwogroups.Rule:supposeeachtrialofanexperimentcanresultinSorF,butthesamplingiswithoutreplacementfromapopulationofsizeN.Ifthesamplesize(numberoftrials)nisatmost5%ofthepopulationsize,theexperimentcanbeanalyzedasthoughitwereexactlyabinomialexperiment.TheBinomialRandomVariableAndDistributionDefinition:Givenabinomialexperimentconsistingofntrials,thebinomialrandomvariableXassociatedwiththisexperimentisdefinedasX=thenumberofS’samongthentrialsThebinomialdistributionisthedistributionofnumberofsuccessesXinnindependentsuccess/failexperimentswithsuccessprobabilityp.WedenotethedistributionasBin(n,p).

IfXfollowsBin(n,p)distribution,thepossiblevaluesofXare0,1,2,...,n.Ifkisanyoneofthesevalues,

Notation:becausethepmfofabinomialrvXdependsonthetwoparametersnandp,wedenotethepmfbyb(x;n,p)Example:eachofsixrandomlyselectedcoladrinkersisgivenaglasscontainingcolaSandonecontainingcolaF.theglassesareidenticalinappearanceexceptforacodeonthebottomtoidentifythecola.Supposethereisactuallynotendencyamongcoladrinkerstopreferonecolatotheother.thenp=P(aselectedindividualprefersS)=.5,sowithX=thenumberamongthesixwhoreferS,X~Bin(6,.5)thusTheprobabilitythatatleastthreepreferSisTheprobabilitythatatmostonepreferSisUsingBinomialTablesNotation:forX~Bin(n,p),thecdfwillbedenotedbyExample:supposethat20%ofallcopiesofaparticulartextbookfailacertainbindingstrengthtest.Letxdenotethenumberamong15randomlyselectedcopiesthatfailthetest.ThenXhasabinomialdistributionwithn=15andp=0.2.1.Theprobabilitythatatmost8failthetestis2.Theprobabilitythatexactly8failis3.Theprobabilitythatatleast8failis4.Theprobabilitythatbetween4and7failisIfwerepeataBernoullitrialntimesandrecordthetotalnumberofsuccessesinthesenindependentandidenticaltrials,thenthetotalnumberofsuccesses

followstheBinomialdistribution.LetY=X1+X2+...+Xn

whereX1,X2,...,XnareindependentBernoullitrialswithprobabilityofsuccess=p,thenY~Bin(n,p)

Theprobabilitydistributionischaracterizedbytwoparameters,thesamplesizenandtheprobabilityofsuccessp.Let'sfamiliarizetheBinomialdistributionthroughsomeexamples:Itcanbeseenthatthelargeristhesamplesizen,thesmootheristhecurve.Whennislargeenough,itisgoodenoughtobeproximatedbyaNormaldistribution.MeanE(X)=npIneachBernoullitrial,thereareonly2possibleoutcomes,successandfailure.SupposeweuseanindicatorItorepresentthenumberofsuccessineachtrial.Ifitisasuccess,thenwehavegot1success;ifitisafailure,thenwehavegot0success.ThentheexpectednumberofsuccessineachtrialisE(I)=P(I=1)(1)+P(I=0)(0)

=p(1)+q(0)

=pThemeanandvarianceofXIfX~Bin(n,p)E(X)istheexpectednumberofsuccessesinnindependenttrials.Itcanbesimplycomputedasthesumoftheexpectedsuccessineachtrialoverntrials,E(X)=E(I)+E(I)+...+E(I)

=p+p+...+p

=np

VarianceVar(X)=np(1-p)Similary,thevarianceoftheindicatorIis

hencethevarianceofX,thenumberofsuccessesinntrials,isthesumofnindependentvarianceofI,Var(X)=Var(I)+Var(I)+...+Var(I)

=pq+pq+...+pq

=npq

=np(1-p)WecanalsocomputetheexpectationandvarianceofX

theoretically.As

Xhastheprobabilitymassfunction,SotheexpectationofXisAndthevarianceofXiswhere

Combiningwehave3.6ThePoissonProbabilityDistributionAdiscreterandomvariableissaidtofollowaPoissondistributionifitsprobabilitymassfunctionisinformof

where

>0.ThePoissonvariableisdenotedbyX~P(

),where

isthemeanofthevariableX.Definition:Thevalueofisfrequentlyarateperunittimeorperunitarea.TheprobabilitythatatrapcontainsexactlyfivecreaturesisExampleletXdenotethenumberofcreaturesofaparticulartypecapturedinatrapduringagiventimeperiod.SupposethatXhasaPossiondistributionwith=4.5,soonaveragetrapswillcontain4.5creatures.TheprobabilitythatatraphasatmostfivecreaturesisTosummarizethephysicalpropertiesofaPoissonvariable:HowtorecognizethePoissonDistributionthroughitsphysicalproperties?Poissonvariableistheresultofacollectionofoccurrencesforwhichtheprobabilityofeachoccurrenceisverysmall.QueuingmodelandaccidentdataareparticularexamplesofPoissondistribution.Sim