181
5
30
12=2loavesofbread
TheprobabilityquestionasksyoutofindP(3).
Example4.21
Acertainbankexpectstoreceive6badchecksperday,onaverage.Whatistheprobabilityofthe
bankgettingfewerthan5badchecksonanygivenday? Ofinterestisthenumberofchecksthe
bankreceivesin1day,sothetimeintervalofinterestis1day. LetX=thenumberofbadchecks
thebankreceivesinoneday.Ifthebankexpectstoreceive6badchecksperdaythentheaverage
is6checksperday.TheprobabilityquestionasksforP(x<5).
Example4.22
Younoticethatanewsreportersays"uh",onaverage,2timesperbroadcast.Whatistheprobabil-
itythatthenewsreportersays"uh"morethan2timesperbroadcast.
ThisisaPoissonproblembecauseyouareinterestedinknowingthenumberoftimesthenews
reportersays"uh"duringabroadcast.
Problem1
(Solutiononp.214.)
Whatistheintervalofinterest?
Problem2
(Solutiononp.214.)
Whatistheaveragenumberoftimesthenewsreportersays"uh"duringonebroadcast?
Problem3
(Solutiononp.214.)
LetX=____________.WhatvaluesdoesXtakeon?
Problem4
(Solutiononp.214.)
TheprobabilityquestionisP(______).
4.8.1NotationforthePoisson:P=PoissonProbabilityDistributionFunction
XP(m)
Readthisas"XisarandomvariablewithaPoissondistribution."Theparameterism(orl). m(orl)=the
meanfortheintervalofinterest.
Example4.23
Leah’sansweringmachinereceivesabout6telephonecallsbetween8a.m. and10a.m. . Whatis
theprobabilitythatLeahreceivesmorethan1callinthenext15minutes?
LetX=thenumberofcallsLeahreceivesin15minutes.(Theintervalofinterestis15minutesor
1
4
hour.)
x=0,1,2,3,...
IfLeahreceives,ontheaverage,6telephonecallsin2hours,andthereareeight15minutesinter-
valsin2hours,thenLeahreceives
1
8
6=0.75
callsin15minutes,ontheaverage.So,m=0.75forthisproblem.
XP(0.75)
FindP(x>1).P(x>1)=0.1734(calculatororcomputer)
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182
CHAPTER4. DISCRETERANDOMVARIABLES
TI-83+andTI-84: Forageneraldiscussion,see thisexample(Binomial). Thesyntaxissimilar.
ThePoissonparameterlistis(mfortheintervalofinterest,number).Forthisproblem:
Press1-andthenpress2ndDISTR.ArrowdowntoC:poissoncdf. PressENTER.Enter.75,1).
TheresultisP(x>1)=0.1734.NOTE:TheTIcalculatorsusel(lambda)forthemean.
TheprobabilitythatLeahreceivesmorethan1telephonecallinthenextfifteenminutesisabout
0.1734.
ThegraphofXP(0.75)is:
They-axiscontainstheprobabilityofxwhereX=thenumberofcallsin15minutes.
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183
4.9SummaryofFunctions
9
Formula4.1:Binomial
XB(n,p)
X=thenumberofsuccessesinnindependenttrials
n=thenumberofindependenttrials
Xtakesonthevaluesx=0,1,2,3,...,n
p=theprobabilityofasuccessforanytrial
q=theprobabilityofafailureforanytrial
p+q=1
q=p
Themeanism=np.Thestandarddeviationiss=
p
npq.
Formula4.2:Geometric
XG(p)
X=thenumberofindependenttrialsuntilthefirstsuccess(countthefailuresandthefirstsuccess)
Xtakesonthevaluesx=1,2,3,...
p=theprobabilityofasuccessforanytrial
q=theprobabilityofafailureforanytrial
p+q=1
q=p
Themeanism=
1
p
Thestandarddeviationiss=
r
1
p

1
p
1
Formula4.3:Hypergeometric
XH(r,b,n)
X=thenumberofitemsfromthegroupofinterestthatareinthechosensample.
Xmaytakeonthevaluesx=0,1,...,uptothesizeofthegroupofinterest.(Theminimumvalue
forXmaybelargerthan0insomeinstances.)
r=thesizeofthegroupofinterest(firstgroup)
b=thesizeofthesecondgroup
n=thesizeofthechosensample.
nr+b
Themeanis:m=
nr
r+b
9
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184
CHAPTER4. DISCRETERANDOMVARIABLES
Thestandarddeviationis:s=
r
rbn(r+b n)
(r+b)
2
(r+b 1)
Formula4.4:Poisson
XP(m)
X=thenumberofoccurrencesintheintervalofinterest
Xtakesonthevaluesx=0,1,2,3,...
Themeanmistypicallygiven. (lisoftenusedasthemeaninsteadofm.) ) WhenthePoissonis
usedtoapproximatethebinomial,weusethebinomialmeanm=npnisthebinomialnumber
oftrials. p=theprobabilityofasuccessforeachtrial.Thisformulaisvalidwhennis"large"and
p"small"(ageneralruleisthatnshouldbegreaterthanorequalto20andpshouldbelessthan
orequalto0.05). IfnislargeenoughandpissmallenoughthenthePoissonapproximatesthe
binomialverywell.Thevarianceiss
2
=mandthestandarddeviationiss=
p
m
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185
4.10Practice1:DiscreteDistribution
10
4.10.1StudentLearningOutcomes
 Thestudentwillanalyzethepropertiesofadiscretedistribution.
4.10.2Given:
Aballetinstructorisinterestedinknowingwhatpercentofeachyear’sclasswillcontinueontothenext,
sothatshecanplanwhatclassestooffer. Overtheyears,shehasestablishedthefollowingprobability
distribution.
 LetX=thenumberofyearsastudentwillstudyballetwiththeteacher.
 LetP
(x)=theprobabilitythatastudentwillstudyballetxyears.
4.10.3OrganizetheData
Completethetablebelowusingthedataprovided.
x
P(x)
x*P(x)
1
0.10
2
0.05
3
0.10
4
5
0.30
6
0.20
7
0.10
Table4.9
Exercise4.10.1
Inwords,definetheRandomVariableX.
Exercise4.10.2
P(4) =
Exercise4.10.3
P(x<4)=
Exercise4.10.4
Onaverage,howmanyyearswouldyouexpectachildtostudyballetwiththisteacher?
4.10.4DiscussionQuestion
Exercise4.10.5
Whatdoesthecolumn"P(x)"sumtoandwhy?
Exercise4.10.6
Whatdoesthecolumn"xP(x)"sumtoandwhy?
10
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186
CHAPTER4. DISCRETERANDOMVARIABLES
4.11Practice2:BinomialDistribution
11
4.11.1StudentLearningOutcomes
 ThestudentwillconstructtheBinomialDistribution.
4.11.2Given
The Higher r Education n Research Institute at UCLA collected d data a from 203,967 incoming first-time,
full-timefreshmenfrom270four-yearcollegesanduniversitiesintheU.S.71.3%ofthosestudentsreplied
that, yes, theybelieve e that t same-sex couples should have e the e right to legal l marital status. . (Source:
http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf).)
Suppose that you randomlypick8first-time, , full-time e freshmenfrom m the e survey. . You u are interested
inthenumberthatbelievesthatsamesex-couplesshouldhavetherighttolegalmaritalstatus
4.11.3InterprettheData
Exercise4.11.1
(Solutiononp.214.)
Inwords,definetherandomVariableX.
Exercise4.11.2
(Solutiononp.214.)
X___________
Exercise4.11.3
(Solutiononp.214.)
WhatvaluesdoestherandomvariableXtakeon?
Exercise4.11.4
Constructtheprobabilitydistributionfunction(PDF).
x
P(x)
Table4.10
Exercise4.11.5
(Solutiononp.214.)
Onaverage(u),howmanywouldyouexpecttoansweryes?
Exercise4.11.6
(Solutiononp.214.)
Whatisthestandarddeviation(s)?
11
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187
Exercise4.11.7
(Solutiononp.214.)
Whatistheprobabilitythatatmost5ofthefreshmenreply“yes”?
Exercise4.11.8
(Solutiononp.214.)
Whatistheprobabilitythatatleast2ofthefreshmenreply“yes”?
Exercise4.11.9
Constructahistogramorplotalinegraph. Labelthehorizontalandverticalaxeswithwords.
Includenumericalscaling.
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188
CHAPTER4. DISCRETERANDOMVARIABLES
4.12Practice3:PoissonDistribution
12
4.12.1StudentLearningOutcomes
 ThestudentwillanalyzethepropertiesofaPoissondistribution.
4.12.2Given
On average, , eight t teens in the U.S. die from
motor vehicle injuries per day.
As
a result, , states s across the country are debating g raising g the driving age.
(Source:
http://www.cdc.gov/Motorvehiclesafety/Teen_Drivers/teendrivers_factsheet.html))
4.12.3InterprettheData
Exercise4.12.1
Assumetheeventoccursindependentlyinanygivenday.Inwords,definetheRandomVariable
X.
Exercise4.12.2
(Solutiononp.214.)
X______________
Exercise4.12.3
(Solutiononp.214.)
WhatvaluesdoesXtakeon?
Exercise4.12.4
ForthegivenvaluesoftherandomvariableX,fillinthecorrespondingprobabilities.
x
P(x)
0
4
8
10
11
15
Table4.11
Exercise4.12.5
(Solutiononp.214.)
IsitlikelythattherewillbenoteenskilledintheU.S.frommotorvehicleinjuriesonanygiven
day?Justifyyouranswernumerically.
Exercise4.12.6
(Solutiononp.214.)
Isitlikelythattherewillbemorethan20teenskilledintheU.S.frommotorvehicleinjurieson
anygivenday?Justifyyouranswernumerically.
12
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189
4.13Practice4:GeometricDistribution
13
4.13.1StudentLearningOutcomes
 Thestudentwillanalyzethepropertiesofageometricdistribution.
4.13.2Given:
UsetheinformationfromtheBinomialDistributionPractice(Section4.11)shownbelow.
The Higher r Education n Research Institute at UCLA collected d data a from 203,967 incoming first-time,
full-timefreshmenfrom270four-yearcollegesanduniversitiesintheU.S.71.3%ofthosestudentsreplied
that, yes, theybelieve e that t same-sex couples should have e the e right to legal l marital status. . (Source:
http://heri.ucla.edu/PDFs/pubs/TFS/Norms/Monographs/TheAmericanFreshman2011.pdf)
Suppose that you randomly select freshman n from the study y until l you find one e who o replies s “yes.”
Youareinterestedinthenumberoffreshmenyoumustask.
4.13.3InterprettheData
Exercise4.13.1
Inwords,definetheRandomVariableX.
Exercise4.13.2
(Solutiononp.215.)
X
Exercise4.13.3
(Solutiononp.215.)
WhatvaluesdoestherandomvariableXtakeon?
Exercise4.13.4
Constructtheprobabilitydistributionfunction(PDF).Stopatx=6.
x
P(x)
1
2
3
4
5
6
Table4.12
Exercise4.13.5
(Solutiononp.215.)
Onaverage(m),howmanyfreshmenwouldyouexpecttohavetoaskuntilyoufoundonewho
replies"yes?"
Exercise4.13.6
(Solutiononp.215.)
Whatistheprobabilitythatyouwillneedtoaskfewerthan3freshmen?
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190
CHAPTER4. DISCRETERANDOMVARIABLES
Exercise4.13.7
Constructahistogramorplotalinegraph. Labelthehorizontalandverticalaxeswithwords.
Includenumericalscaling.
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