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b. Identifythefollowingvalues:
i. Lowestvalueforx:
ii. Highestvalueforx:
iii. Heightoftherectangle:
iv. Labelforx-axis(words):
v. Labelfory-axis(words):
5.6.5RandomProbability
Exercise5.6.8
(Solutiononp.257.)
Findtheprobabilitythatarandomlychosencarinthelotwaslessthan4yearsold.
a. Sketchthegraph.Shadetheareaofinterest.
Figure5.9
b. Findtheprobability.P(x<4)=
Exercise5.6.9
(Solutiononp.257.)
Outofjustthecarslessthan7.5yearsold,findtheprobabilitythatarandomlychosencarinthe
lotwaslessthan4yearsold.
a. Sketchthegraph.Shadetheareaofinterest.
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CHAPTER5. CONTINUOUSRANDOMVARIABLES
Figure5.10
b. Findtheprobability.P(x<4jx<7.5)=
Exercise5.6.10:DiscussionQuestion
Whathaschangedintheprevioustwoproblemsthatmadethesolutionsdifferent?
5.6.6Quartiles
Exercise5.6.11
(Solutiononp.257.)
Findtheaverageageofthecarsinthelot.
Exercise5.6.12
(Solutiononp.257.)
Findthethirdquartileofagesofcarsinthelot. Thismeansyouwillhavetofindthevaluesuch
that
3
4
,or75%,ofthecarsareatmost(lessthanorequalto)thatage.
a. Sketchthegraph.Shadetheareaofinterest.
Figure5.11
b. FindthevalueksuchthatP(x<k)=0.75.
c. Thethirdquartileis:
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243
5.7Practice2:ExponentialDistribution
7
5.7.1StudentLearningOutcomes
 Thestudentwillanalyzedatafollowingtheexponentialdistribution.
5.7.2Given
Carbon-14isaradioactiveelementwithahalf-lifeofabout5730years.Carbon-14issaidtodecayexponen-
tially. Thedecayrateis0.000121.Westartwith1gramofcarbon-14.Weareinterestedinthetime(years)
ittakestodecaycarbon-14.
5.7.3DescribetheData
Exercise5.7.1
Whatisbeingmeasuredhere?
Exercise5.7.2
(Solutiononp.258.)
Arethedatadiscreteorcontinuous?
Exercise5.7.3
(Solutiononp.258.)
Inwords,definetheRandomVariableX.
Exercise5.7.4
(Solutiononp.258.)
Whatisthedecayrate(m)?
Exercise5.7.5
(Solutiononp.258.)
ThedistributionforXis:
5.7.4Probability
Exercise5.7.6
(Solutiononp.258.)
Findtheamount(percentof1gram)ofcarbon-14lastinglessthan5730years. Thismeans,find
P(x<5730).
a. Sketchthegraph.Shadetheareaofinterest.
Figure5.12
7
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CHAPTER5. CONTINUOUSRANDOMVARIABLES
b. Findtheprobability.P(x<5730)=
Exercise5.7.7
(Solutiononp.258.)
Findthepercentageofcarbon-14lastinglongerthan10,000years.
a. Sketchthegraph.Shadetheareaofinterest.
Figure5.13
b. Findtheprobability.P(x>10000)=
Exercise5.7.8
(Solutiononp.258.)
Thirtypercent(30%)ofcarbon-14willdecaywithinhowmanyyears?
a. Sketchthegraph.Shadetheareaofinterest.
Figure5.14
b. FindthevalueksuchthatP(x<k)=0.30.
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5.8Homework
8
Foreachprobabilityandpercentileproblem,DRAWTHEPICTURE!
Exercise5.8.1
Considerthefollowingexperiment. Youareoneof100peopleenlistedtotakepartinastudyto
determinethepercentofnursesinAmericawithanR.N.(registerednurse)degree.Youasknurses
iftheyhaveanR.N.degree.Thenursesanswer“yes”or“no.”Youthencalculatethepercentage
ofnurseswithanR.N.degree.Yougivethatpercentagetoyoursupervisor.
a. Whatpartoftheexperimentwillyielddiscretedata?
b. Whatpartoftheexperimentwillyieldcontinuousdata?
Exercise5.8.2
Whenageisroundedtothenearestyear,dothedatastaycontinuous,ordotheybecomediscrete?
Why?
Exercise5.8.3
(Solutiononp.258.)
Birthsareapproximatelyuniformlydistributedbetweenthe52weeksoftheyear. Theycanbe
saidtofollowaUniformDistributionfrom1–53(spreadof52weeks).
a. X
b. Graphtheprobabilitydistribution.
c. f(x)=
d. m=
e. s=
f. Findtheprobabilitythat apersonisbornat theexactmomentweek19starts. . That t is, find
P(x=19)=
g. P(2<x<31)=
h. Findtheprobabilitythatapersonisbornafterweek40.
i. P(12<xjx<28)=
j. Findthe70thpercentile.
k. Findtheminimumfortheupperquarter.
Exercise5.8.4
Arandomnumbergeneratorpicksanumberfrom1to9inauniformmanner.
a. X~
b. Graphtheprobabilitydistribution.
c. f(x)=
d. m=
e. s=
f. P(3.5<x<7.25)=
g. P(x>5.67)=
h. P(x>5jx>3)=
i. Findthe90thpercentile.
Exercise5.8.5
(Solutiononp.258.)
Thetime(inminutes)untilthenextbusdepartsamajorbusdepotfollowsadistributionwith
f(x)=
1
20
wherexgoesfrom25to45minutes.
a. Definetherandomvariable.X=
8
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CHAPTER5. CONTINUOUSRANDOMVARIABLES
b. X~
c. Graphtheprobabilitydistribution.
d. Thedistributionis______________(nameofdistribution).Itis_____________(discreteorcon-
tinuous).
e. m=
f. s=
g. Findtheprobabilitythatthetimeisatmost30minutes.Sketchandlabelagraphofthedistri-
bution.Shadetheareaofinterest.Writetheanswerinaprobabilitystatement.
h. Findtheprobabilitythatthetimeisbetween30and40minutes. Sketchandlabelagraphof
thedistribution.Shadetheareaofinterest.Writetheanswerinaprobabilitystatement.
i. P(25<x<55)=_________. Statethisinaprobabilitystatement(similartogandh),draw
thepicture,andfindtheprobability.
j. Findthe90thpercentile.Thismeansthat90%ofthetime,thetimeislessthan_____minutes.
k. Findthe75thpercentile.Inacompletesentence,statewhatthismeans.(Seej.)
l. Findtheprobabilitythatthetimeismorethan40minutesgiven(orknowingthat)itisatleast
30minutes.
Exercise5.8.6
AccordingtoastudybyDr. JohnMcDougallofhislive-inweightlossprogramatSt. Helena
Hospital,thepeoplewhofollowhisprogramlosebetween6and15poundsamonthuntilthey
approachtrimbodyweight. Let’ssupposethattheweightlossisuniformlydistributed. . Weare
interestedinthe weightlossofarandomlyselectedindividualfollowingtheprogramforone
month. (Source: TheMcDougallProgramforMaximumWeightLossbyJohnA.McDougall,
M.D.)
a. Definetherandomvariable.X=
b. X~
c. Graphtheprobabilitydistribution.
d. f(x)=
e. m=
f. s=
g. Findtheprobabilitythattheindividuallostmorethan10poundsinamonth.
h. Supposeitisknownthattheindividuallostmorethan10poundsinamonth.Findtheproba-
bilitythathelostlessthan12poundsinthemonth.
i. P(7<x<13jx>9)=__________. Statethisinaprobabilityquestion(similartogandh),
drawthepicture,andfindtheprobability.
Exercise5.8.7
(Solutiononp.258.)
AsubwaytrainontheRedLinearrivesevery8minutesduringrushhour.Weareinterestedinthe
lengthoftimeacommutermustwaitforatraintoarrive.Thetimefollowsauniformdistribution.
a. Definetherandomvariable.X=
b. X~
c. Graphtheprobabilitydistribution.
d. f(x)=
e. m=
f. s=
g. Findtheprobabilitythatthecommuterwaitslessthanoneminute.
h. Findtheprobabilitythatthecommuterwaitsbetweenthreeandfourminutes.
i. 60%ofcommuterswaitmorethanhowlongforthetrain? Statethisinaprobabilityquestion
(similartogandh),drawthepicture,andfindtheprobability.
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Exercise5.8.8
TheageofafirstgraderonSeptember1atGardenElementarySchoolisuniformlydistributed
from5.8to6.8years.Werandomlyselectonefirstgraderfromtheclass.
a. Definetherandomvariable.X=
b. X~
c. Graphtheprobabilitydistribution.
d. f(x)=
e. m=
f. s=
g. Findtheprobabilitythatsheisover6.5years.
h. Findtheprobabilitythatsheisbetween4and6years.
i. Findthe70thpercentilefortheageoffirstgradersonSeptember1atGardenElementarySchool.
Exercise5.8.9
(Solutiononp.259.)
LetX~Exp(0.1)
a. decayrate=
b. m=
c. Graphtheprobabilitydistributionfunction.
d. Ontheabovegraph,shadetheareacorrespondingtoP(x<6)andfindtheprobability.
e. Sketchanewgraph,shadetheareacorrespondingtoP(3<x<6)andfindtheprobability.
f. Sketchanewgraph,shadetheareacorrespondingtoP(x>7)andfindtheprobability.
g. Sketchanewgraph,shadetheareacorrespondingtothe40thpercentileandfindthevalue.
h. Findtheaveragevalueofx.
Exercise5.8.10
Supposethatthelengthoflongdistancephonecalls,measuredinminutes,isknowntohavean
exponentialdistributionwiththeaveragelengthofacallequalto8minutes.
a. Definetherandomvariable.X=
b. IsXcontinuousordiscrete?
c. X~
d. m=
e. s=
f. Drawagraphoftheprobabilitydistribution.Labeltheaxes.
g. Findtheprobabilitythataphonecalllastslessthan9minutes.
h. Findtheprobabilitythataphonecalllastsmorethan9minutes.
i. Findtheprobabilitythataphonecalllastsbetween7and9minutes.
j. If25phonecallsaremadeoneafteranother,onaverage,whatwouldyouexpectthetotaltobe?
Why?
Exercise5.8.11
(Solutiononp.259.)
Supposethattheusefullifeofaparticularcarbattery,measuredinmonths,decayswithparameter
0.025.Weareinterestedinthelifeofthebattery.
a. Definetherandomvariable.X=
b. IsXcontinuousordiscrete?
c. X~
d. Onaverage,howlongwouldyouexpect1carbatterytolast?
e. Onaverage,howlongwouldyouexpect9carbatteriestolast,iftheyareusedoneafteranother?
f. Findtheprobabilitythatacarbatterylastsmorethan36months.
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CHAPTER5. CONTINUOUSRANDOMVARIABLES
g. 70%ofthebatterieslastatleasthowlong?
Exercise5.8.12
Thepercentofpersons(ages5andolder)ineachstatewhospeakalanguageathomeotherthan
Englishisapproximatelyexponentiallydistributedwithameanof9.848.Supposewerandomly
pickastate.(Source:BureauoftheCensus,U.S.Dept.ofCommerce)
a. Definetherandomvariable.X=
b. IsXcontinuousordiscrete?
c. X~
d. m=
e. s=
f. Drawagraphoftheprobabilitydistribution.Labeltheaxes.
g. Findtheprobabilitythatthepercentislessthan12.
h. Findtheprobabilitythatthepercentisbetween8and14.
i. ThepercentofallindividualslivingintheUnitedStateswhospeakalanguageathomeother
thanEnglishis13.8.
i. Whyisthisnumberdifferentfrom9.848%?
ii. Whatwouldmakethisnumberhigherthan9.848%?
Exercise5.8.13
(Solutiononp.259.)
The time (inyears)afterreachingage60thatittakesanindividualtoretireisapproximately
exponentiallydistributedwithameanofabout5years. Supposewerandomlypickoneretired
individual.Weareinterestedinthetimeafterage60toretirement.
a. Definetherandomvariable.X=
b. IsXcontinuousordiscrete?
c. X~
d. m=
e. s=
f. Drawagraphoftheprobabilitydistribution.Labeltheaxes.
g. Findtheprobabilitythatthepersonretiredafterage70.
h. Domorepeopleretirebeforeage65orafterage65?
i. Inaroomof1000peopleoverage80,howmanydoyouexpectwillNOThaveretiredyet?
Exercise5.8.14
Thecostofallmaintenance foracar duringitsfirst yearisapproximatelyexponentiallydis-
tributedwithameanof$150.
a. Definetherandomvariable.X=
b. X~
c. m=
d. s=
e. Drawagraphoftheprobabilitydistribution.Labeltheaxes.
f. Findtheprobabilitythatacarrequiredover$300formaintenanceduringitsfirstyear.
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249
5.8.1Trythesemultiplechoiceproblems
Thenextthreequestionsrefertothefollowinginformation. Theaveragelifetimeofacertainnewcell
phoneis3years.Themanufacturerwillreplaceanycellphonefailingwithin2yearsofthedateofpurchase.
Thelifetimeofthesecellphonesisknowntofollowanexponentialdistribution.
Exercise5.8.15
(Solutiononp.259.)
Thedecayrateis
A. 0.3333
B. 0.5000
C. 2.0000
D. 3.0000
Exercise5.8.16
(Solutiononp.259.)
Whatistheprobabilitythataphonewillfailwithin2yearsofthedateofpurchase?
A. 0.8647
B. 0.4866
C. 0.2212
d. 0.9997
Exercise5.8.17
(Solutiononp.259.)
Whatisthemedianlifetimeofthesephones(inyears)?
A. 0.1941
B. 1.3863
C. 2.0794
D. 5.5452
Thenextthreequestionsrefertothefollowinginformation.TheSkyTrainfromtheterminaltotherental
carandlongtermparkingcenterissupposedtoarriveevery8minutes.Thewaitingtimesforthetrainare
knowntofollowauniformdistribution.
Exercise5.8.18
(Solutiononp.259.)
Whatistheaveragewaitingtime(inminutes)?
A. 0.0000
B. 2.0000
C. 3.0000
D. 4.0000
Exercise5.8.19
(Solutiononp.259.)
Findthe30thpercentileforthewaitingtimes(inminutes).
A. 2.0000
B. 2.4000
C. 2.750
D. 3.000
Exercise5.8.20
(Solutiononp.259.)
Theprobabilityofwaitingmorethan7minutesgivenapersonhaswaitedmorethan4minutes
is?
A. 0.1250
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CHAPTER5. CONTINUOUSRANDOMVARIABLES
B. 0.2500
C. 0.5000
D. 0.7500
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