Chapter3.RandomVariablesandProbabilityDistribution

ConceptofaRandomVariable

Example:threeelectroniccomponentsaretested

samplespace(N:nondefective,D:defective)

S=｛NNN,NND,NDN,DNN,NDD,DND,DDN,DDD｝

allocateanumericaldescriptionofeachoutcome

concernedwiththenumberofdefectives

eachpointinthesamplespacewillbeassignedanumericalvalueof0,1,2,or3.

randomvariableX:thenumberofdefectiveitems,arandomquantity

randomvariable

Definition3.1

Arandomvariableisafunctionthatassociatesarealnumberwitheachelementinthesamplespace.

X:arandomvariable

x:oneofitsvalues

EachpossiblevalueofXrepresentsaneventthatisasubsetofthesamplespace

electroniccomponenttest:

E=｛DDN,DND,NDD｝=｛X=2｝.

Example3.1Twoballsaredrawninsuccessionwithoutreplacementfromanurncontaining4redballsand3blackballs.Yisthenumberofredballs.ThepossibleoutcomesandthevaluesyoftherandomvariableY?

Example3.2Astockroomclerkreturnsthreesafetyhelmetsatrandomtothreesteelmillemployeeswhohadpreviouslycheckedthem.IfSmith,Jones,andBrown,inthatorder,receiveoneofthethreehats,listthesamplepointsforthepossibleordersofreturningthehelmets,andfindthevaluemoftherandomvariableMthatrepresentsthenumberofcorrectmatches.

ThesamplespacecontainsafinitenumberofelementsinExample3.1and3.2.

anotherexample:adieisthrownuntila5occurs,

F:theoccurrenceofa5

N:thenonoccurrenceofa5

obtainasamplespacewithanunendingsequenceofelements

S=｛F,NF,NNF,NNNF,...｝

thenumberofelementscanbeequatedtothenumberofwholenumbers;canbecounted

Definition3.2Ifasamplespacecontainsafinitenumberofpossibilitiesoranunendingsequencewithasmanyelementsastherearewholenumbers,itiscalledadiscretesamplespace.

Theoutcomesofsomestatisticalexperimentsmaybeneitherfinitenorcountable.

example:measurethedistancesthatacertainmakeofautomobilewilltraveloveraprescribedtestcourseon5litersofgasoline

distance:avariablemeasuredtoanydegreeofaccuracy

wehaveinfinitenumberofpossibledistancesinthesamplespace,cannotbeequatedtothenumberofwholenumbers.

Definition3.3

Ifasamplespacecontainsaninfinitenumberofpossibilitiesequaltothenumberofpointsonalinesegment,itiscalledacontinuoussamplespace

Arandomvariableiscalledadiscreterandomvariableifitssetofpossibleoutcomesiscountable.

YinExample3.1andMinExample3.2arediscreterandomvariables.

Whenarandomvariablecantakeonvaluesonacontinuousscale,itiscalledacontinuousrandomvariable.

Themeasureddistancethatacertainmakeofautomobilewilltraveloveratestcourseon5litersofgasolineisacontinuousrandomvariable.

continuousrandomvariablesrepresentmeasureddata:

allpossibleheights,weights,temperatures,distance,orlifeperiods.

discreterandomvariablesrepresentcountdata:thenumberofdefectivesinasampleofkitems,orthenumberofhighwayfatalitiesperyearinagivenstate.

2.DiscreteProbabilityDistribution

Adiscreterandomvariableassumeseachofitsvalueswithacertainprobability

assumeequalweightsfortheelementsinExample3.2,what'stheprobabilitythatnoemployeegetsbackhisrighthelmet.

TheprobabilitythatMassumedthevaluezerois1/3.

ThepossiblevaluesmofMandtheirprobabilitiesare

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ProbabilityMassFunction

ItisconvenienttorepresentalltheprobabilitiesofarandomvariableXbyaformula.

writep(x)=P(X=x)

Thesetoforderedpairs(x,p(x))iscalledtheprobabilityfunctionorprobabilitydistributionofthediscreterandomvariableX.

Definition3.4

Thesetoforderedpairs(x,p(x))isaprobabilityfunction,probabilitymassfunction,orprobabilitydistributionofthediscreterandomvariableXif,foreachpossibleoutcomex

Example3.3Ashipmentof8similarmicrocomputerstoaretailoutletcontains3thataredefective.Ifaschoolmakesarandompurchaseof2ofthesecomputers,findtheprobabilitydistributionforthenumberofdefectives.

Solution

X:thepossiblenumbersofdefectivecomputers

xcanbeanyofthenumbers0,1,and2.

CumulativeFunction

TherearemanyproblemwherewemaywishtocomputetheprobabilitythattheobservedvalueofarandomvariableXwillbelessthanorequaltosomerealnumberx.

WritingF(x)=P(X≤x)foreveryrealnumberx.

Definition3.5

ThecumulativedistributionF(x)ofadiscreterandomvariableXwithprobabilitydistributionp(x)is

FortherandomvariableM,thenumberofcorrectmatchesinExample3.2,wehave

ThecumulativedistributionofMis

Remark.thecumulativedistributionisdefinednotonlyforthevaluesassumedbygivenrandomvariablebutforallrealnumbers.

Example3.5TheprobabilitydistributionofXis

FindthecumulativedistributionoftherandomvariableX.

Certainprobabilitydistributionareapplicabletomorethanonephysicalsituation.

TheprobabilitydistributionofExample3.5canapplytodifferentexperimentalsituations.

Example1:thedistributionofY,thenumberofheadswhenacoinistossed4times

Example2:thedistributionofW,thenumberofreadcardsthatoccurwhen4cardsaredrawnatrandomfromadeckinsuccessionwitheachcardreplacedandthedeckshuffledbeforethenextdrawing.

graphs

Itishelpfultolookataprobabilitydistributioningraphicform.

barchart;

histogram;

cumulativedistribution.

ContinuousProbabilityDistribution

ContinuousProbabilitydistribution

Acontinuousrandomvariablehasaprobabilityofzeroofassumingexactlyanyofitsvalues.Consequently,itsprobabilitydistributioncannotbegivenintabularform.

Example:theheightsofallpeopleover21yearsofage(randomvariable)

Between163.5and164.5centimeters,oreven163.99and164.01centimeters,thereareaninfinitenumberofheights,oneofwhichis164centimeters.

Theprobabilityofselectingapersonatrandomwhoisexactly164centimeterstallandnotoneoftheinfinitelylargesetofheightssocloseto164centimetersisremote.

Weassignaprobabilityofzerotoapoint,butthisisnotthecaseforaninterval.Wewilldealwithanintervalratherthanapointvalue,suchasP(a<X<b),P(W≥c).

P(a≤X≤b)=P(a<X≤b)=P(a≤X<b)=P(a<X<b)

whereXiscontinuous.Itdoesnotmatterwhetherweincludeanendpointoftheintervalornot.ThisisnottruewhenXisdiscrete.

Althoughtheprobabilitydistributionofacontinuousrandomvariablecannotbepresentedintabularform,itcanbestatedasaformula.

refertohistogram

Definition3.6Thefunctionf(x)isaprobabilitydensityfunctionforthecontinuousrandomvariableX,definedoverthesetofrealnumbersR,if

Example3.6Supposethattheerrorinthereactiontemperature,inoC,foracontrolledlaboratoryexperimentisacontinuousrandomvariableXhavingtheprobabilitydensityfunction

(a)Verifycondition2ofDefinition3.6.

(b)FindP(0<X≤1).

Solution:......P(0<X≤1)=1/9.

Definition3.7ThecumulativedistributionF(x)ofacontinuousrandomvariableXwithdensityfunctionf(x)is

immediateconsequence:

Example3.7ForthedensityfunctionofExample3.6find

F(x),anduseittoevaluateP(0<x≤1).

4.JointProbabilityDistributions

theprecedingsections:one-dimensionalsamplespacesandasinglerandomvariable

situations:desirabletorecordthesimultaneousoutcomesofseveralrandomvariables.

JointProbabilityDistribution

Examples

1.wemightmeasuretheamountofprecipitatePandvolumeVofgasreleasedfromacontrolledchemicalexperiment;wegetatwo-dimensionalsamplespaceconsistingoftheoutcomes(p,v).

2.Inastudytodeterminethelikelihoodofsuccessincollege,basedonhighschooldata,onemightuseathree-dimensionalsamplespaceandrecordforeachindividualhisorheraptitudetestscore,highschoolrankinclass,andgrade-pointaverageattheendofthefreshmanyearincollege.

XandYaretwodiscreterandomvariables,thejointprobabilitydistributionofXandYis

p(x,y)=P(X=x,Y=y)

thevaluesp(x,y)givetheprobabilitythatoutcomesxandyoccuratthesametime.

Definition3.8Thefunctionp(x,y)isajointprobabilitydistributionorprobabilitymassfunctionofthediscreterandomvariablesXandYif

Example3.8

Tworefillsforaballpointpenareselectedatrandomfromaboxthatcontains3bluerefills,2redrefills,and3greenrefills.IfXisthenumberofbluerefillsandYisthenumberofredrefillsselected,find

(a)thejointprobabilityfunctionp(x,y)

(b)P[(X,Y)∈A]whereAistheregion｛(x,y)|x+y≤1｝

Solution

thepossiblepairsofvalues(x,y)are(0,0),(0,1),(1,0),(1,1),(0,2),and(2,0).

p(x,y)representstheprobabilitythatxblueandyredrefillsareselected.

(b)P[(X,Y)∈A]=9/14

presenttheresultsinTable3.1

Definition3.9Thefunctionf(x,y)isajointdensityfunctionofthecontinuousrandomvariablesXandYif

WhenXandYarecontinuousrandomvariables,thejointdensityfunctionf(x,y)isasurfacelyingabovethexyplane.

P[(X,Y)∈A],whereAisanyregioninthexyplane,isequaltothevolumeoftherightcylinderboundedbythebaseAandthesurface.

Example3.9Supposethatthejointdensityfunctionis

(b)P[(X,Y)∈A]=13/160

marginaldistribution

p(x,y):thejointprobabilitydistributionofthediscreterandomvariablesXandY

theprobabilitydistributionpX(x)ofXaloneisobtainedbysummingp(x,y)overthevaluesofY.

Similarly,theprobabilitydistributionpY(y)ofYaloneisobtainedbysummingp(x,y)overthevaluesofX.

pX(x)andpY(y):marginaldistributionsofXandY

WhenXandYarecontinuousrandomvariables,summationsarereplacedbyintegrals.

Definition3.10ThemarginaldistributionofXaloneandofYaloneare

Example3.10ShowthatthecolumnandrowtotalsofTable

3.1givethemarginaldistributionofXaloneandofYalone.

Example3.11Findmarginalprobabilitydensityfunctions

fX(x)andfy(y)forthejointdensityfunctionofExample3.9.

ThemarginaldistributionpX(x)[orfX(x)]andpx(y)[orfy(y)]areindeedtheprobabilitydistributionoftheindividualvariableXandY,respectively.

Howtoverify?

TheconditionsofDefinition3.4[orDefinition3.6]aresatisfied.

Conditionaldistribution

recallthedefinitionofconditionalprobability:

XandYarediscreterandomvariables,wehave

Thevaluexoftherandomvariablerepresentaneventthatisasubsetofthesamplespace.

Definition3.11

LetXandYbetwodiscreterandomvariables.TheconditionalprobabilitymassfunctionoftherandomvariableY,giventhatX=x,is

Similarly,theconditionalprobabilitymassfunctionoftherandomvariableX,giventhatY=y,is

Definition3.11'

LetXandYbetwocontinuousrandomvariables.TheconditionalprobabilitydensityfunctionoftherandomvariableY,giventhatX=x,is

Similarly,theconditionalprobabilitydensityfunctionoftherandomvariableX,giventhatY=y,is

Remark:

Thefunctionf(x,y)/fX(x)isstrictlyafunctionofywithxfixed,thefunctionf(x,y)/fy(y)isstrictlyafunctionofxwithyfixed,bothsatisfyalltheconditionsofaprobabilitydistribution.

HowtofindtheprobabilitythattherandomvariableXfallsbetweenaandbwhenitisknownthatY=y

Example3.12ReferringtoExample3.8,findtheconditionaldistributionofX,giventhatY=1,anduseittodetermine

P(X=0|Y=1).

Example3.13Thejointdensityfortherandomvariables(X,Y)whereXistheunittemperaturechangeandYistheproportionofspectrumshiftthatacertainatomicparticleproducesis

FindthemarginaldensitiesfX(x),fy(y),andtheconditionaldensityfYX(y|x)

(b)Findtheprobabilitythatthespectrumshiftsmorethanhalfofthetotalobservations,giventhetemperatureisincreasedto0.25unit.

(a)

(b)

Example3.14Giventhejointdensityfunction

(a)

(b)

statisticalindependence

eventsAandBareindependent,if

P(B∩A)=P(A)P(B).

discreterandomvariablesXandYareindependent,if

P(X=x,Y=y)=P(X=x)P(Y=y)

forall(x,y)withintheirrange.

Thevaluexoftherandomvariablerepresentaneventthatisasubsetofthesamplespace.

Definition3.12LetXandYbetwodiscreterandomvariables,withjointprobabilitydistributionp(x,y)andmarginaldistributionspX(x)andpY(y),respectively.TherandomvariablesXandYaresaidtobestatisticallyindependentifandonlyif

p(x,y)=pX(x)pY(y)forall(x,y)withintheirrange.

Definition3.12'LetXandYbetwocontinuousrandomvariables,withjointprobabilitydistributionf(x,y)andmarginaldistributionsfX(x)andfY(y),respectively.TherandomvariablesXan