Fiddle Faddle« & Screaming Yellow Zonkers«, przepisy, Przepisy Pizza Hut KFC MC DONALDS, Przepisy po angielsku
Fatburger«, przepisy, Przepisy Pizza Hut KFC MC DONALDS, Przepisy po angielsku
Food 20 adjectives, ANGIELSKI, !!!!!!!!!!pomoce, słownictwo, food
Frommer's Naples and the Amalfi Coast day BY Day, Travel Guides- Przewodniki (thanx angielski i stuff)
Frommer s Sicily Day By Day, Travel Guides- Przewodniki (thanx angielski i stuff)
Fuyumi Ono - Twelve Kingdoms 01 - Shadow of the Moon a Sea of Shadows, Angielskie [EN](4)(2)
Farago&Zwijnenberg (eds) - Compelling Visuality ~ The work of art in and out of history, sztuka i nie tylko po angielsku
Forex Study Book For Successful Foreign Exchange Dealing, gielda walutowa, Angielskie
Fabulous Creatures Mythical Monsters and Animal Power Symbols-A Handbook, Angielski
Farkas-Ukryte rzeczywistości, âşâEZOTERYKA,samouzdrawianie, psychologiaâ
Fundamentals of Statistics - 2e - Chapter07, Angielskie [EN](4)(2)
[ Pobierz całość w formacie PDF ]TheNormalProbability
Distribution
CHAPTER
Outline
7. 1
PropertiesoftheNormalDistribution
7. 2
TheStandardNormalDistribution
7. 3
ApplicationsoftheNormalDistribution
7. 4
AssessingNormality
7. 5
TheNormalApproximationtotheBinomial
ProbabilityDistribution
"
ChapterReview
"
CaseStudy:ATaleofBlood,Chemistry,and
Health(OnCD)
DECISIONS
YouareinterestedinstartingyourownMENSA-typeclub.Toqualify
fortheclub,thepotentialmembermusthaveintelligencethatisinthe
top20%ofallpeople.Youmustdecidethebaselinescorethatallowsan
individualtoqualify.SeetheDecisionsprojectonpage359.
PuttingItAllTogether
InChapter6,weintroduceddiscreteprobabilitydistri-
butionsand,inparticular,thebinomialprobabilitydis-
tribution.Wecomputedprobabilitiesforthisdiscrete
distributionusingitsprobabilitydistributionfunction.
However,wecouldalsodeterminetheprobabilityof
anydiscreterandomvariablefromitsprobabilityhis-
togram.Forexample,thefigureshowstheprobability
histogramforthebinomialrandomvariable
X
with
and
Fromtheprobabilityhistogram,wecansee
Noticethatthewidthofeachrectanglein
theprobabilityhistogramis1.Sincetheareaofarectan-
gleequalsheighttimeswidth,wecanthinkof
P
(1)asthe
areaoftherectanglecorrespondingto Thinking
ofprobabilityinthisfashionmakesthetransitionfrom
computingdiscreteprobabilitiestocontinuousprobabili-
tiesmucheasier.
Inthischapter,wediscusstwocontinuousdistribu-
tions,
theuniformdistribution
and
thenormaldistribution
.
Thegreaterpartofthediscussionwillfocusonthenor-
maldistribution,whichhasmanyusesandapplications.
BinomialProbabilityHistogram;
n
#
5,
p
#
0.35
=
0.35.
0.4
0.3
L
0.31.
0.2
0.1
=
1.
012345
318
Section7.1PropertiesofNormalDistribution
319
7.1
PropertiesoftheNormalDistribution
PreparingforThisSection
Beforegettingstarted,reviewthefollowing:
•
Continuousvariable(Section1.1,p.7)
•
Rulesforadiscreteprobabilitydistribution(Section
6.1,p.287)
•
Z
-score(Section3.4,pp.149–150)
•
TheEmpiricalRule(Section3.2,pp.131–132)
Objectives
Understandtheuniformprobabilitydistribution
Graphanormalcurve
Statethepropertiesofthenormalcurve
Understandtheroleofareainthenormaldensity
function
Understandtherelationbetweenanormalrandom
variableandastandardnormalrandomvariable
UnderstandtheUniformProbability
Distribution
Weillustrateauniformdistributionusinganexample.Usingtheuniformdistri-
butionmakesiteasytoseetherelationbetweenareaandprobability.
EXAMPLE1
IllustratingtheUniformDistribution
Imaginethatafriendofyoursisalwayslate.Lettherandomvariable
X
rep-
resentthetimefromwhenyouaresupposedtomeetyourfrienduntilhe
showsup.Furthersupposethatyourfriendcouldbeontime orup
to30minuteslate withall1-minuteintervalsoftimesbetween
and equallylikely.Thatis,yourfriendisjustaslikelytobe
from3to4minuteslateasheistobe25to26minuteslate.Therandomvari-
able
X
canbeanyvalueintheintervalfrom0to30,thatis, Be-
causeanytwointervalsofequallengthbetween0and30,inclusive,are
equallylikely,therandomvariable
X
issaidtofollowa
uniformprobability
distribution
.
=
30
2
=
30
…
30.
Whenwecomputeprobabilitiesfordiscreterandomvariables,weusually
substitutethevalueoftherandomvariableintoaformula.
Thingsarenotaseasyforcontinuousrandomvariables.Sincethereare
aninfinitenumberofpossibleoutcomesforcontinuousrandomvariables
theprobabilityofobservingaparticularvalueofacontinuousrandomvari-
ableiszero.Forexample,theprobabilitythatyourfriendisexactly
12.9438823minuteslateiszero.Thisresultisbasedonthefactthatclassical
probabilityisfoundbydividingthenumberofwaysaneventcanoccurby
thetotalnumberofpossibilities.Thereisonewaytoobserve12.9438823,
andthereareaninfinitenumberofpossiblevaluesbetween0and30,sowe
getaprobabilitythatiszero.Toresolvethisproblem,wecomputeprobabil-
itiesofcontinuousrandomvariablesoveranintervalofvalues.Forexample,
wemightcomputetheprobabilitythatyourfriendisbetween10and15min-
uteslate.Tofindprobabilitiesforcontinuousrandomvariables,weuse
probabilitydensityfunctions
.
320
Chapter7TheNormalProbabilityDistribution
Definition
A
probabilitydensityfunction
isanequationusedtocompute
probabilitiesofcontinuousrandomvariablesthatmustsatisfythefollowing
twoproperties.
1.
Thetotalareaunderthegraphoftheequationoverallpossiblevaluesof
therandomvariablemustequal1.
2.
Theheightofthegraphoftheequationmustbegreaterthanorequalto
0forallpossiblevaluesoftherandomvariable.Thatis,thegraphofthe
equationmustlieonorabovethehorizontalaxisforallpossiblevalues
oftherandomvariable.
InOtherWords
Tofindprobabilitiesforcontinuous
randomvariables,wedonotuse
probabilitydistributionfunctions(aswe
didfordiscreterandomvariables).
Instead,weuseprobabilitydensity
functions.Theword
density
isused
becauseitreferstothenumberof
individualsperunitofarea.
Property1issimilartotherulefordiscreteprobabilitydistributionsthat
statedthesumoftheprobabilitiesmustaddupto1.Property2issimilartothe
rulethatstatedallprobabilitiesmustbegreaterthanorequalto0.
Figure1illustratesthepropertiesfortheexampleaboutyourfriendwhois
alwayslate.Sinceallpossiblevaluesoftherandomvariablebetween0and30
areequallylikely,thegraphoftheprobabilitydensityfunctionforuniformran-
domvariablesisarectangle.Becausetherandomvariableisanynumberbe-
tween0and30inclusive,thewidthoftherectangleis30.Sincetheareaunder
thegraphoftheprobabilitydensityfunctionmustequal1,andtheareaofarec-
tangleequalsheighttimeswidth,theheightoftherectanglemustbe
1
30
.
Figure1
UniformDensityFunction
1
30
Area
$
30
X
RandomVariable(time)
Apressingquestionremains:Howdoweusedensityfunctionstofindprob-
abilitiesofcontinuousrandomvariables?
Theareaunderthegraphofadensityfunctionoversomeintervalrepresents
theprobabilityofobservingavalueoftherandomvariableinthatinterval.
Thefollowingexampleillustratesthisstatement.
EXAMPLE2
AreaasaProbability
Problem
:
RefertothesituationpresentedinExample1.Whatistheproba-
bilitythatyourfriendwillbebetween10and20minuteslatethenexttimeyou
meethim?
Approach
:
Figure1presentedthegraphofthedensityfunction.Weneedto
findtheareaunderthegraphbetween10and20minutes.
Solution
:
Figure2presentsthegraphofthedensityfunctionwiththeareawe
wishtofindshadedingreen.
Section7.1PropertiesoftheNormalDistribution
321
Figure2
1
30
0
10
20
3 0
1
30
Thewidthoftherectangleis10anditsheightis Therefore,thearea
1
30
b
10
a
between10and20is Theprobabilitythatyourfriendisbetween
10and20minuteslateis
NowWorkProblem13.
Weintroducedtheuniformdensityfunctionsothatwecouldassociate
probabilitywitharea.Wearenowbetterpreparedtodiscussthemostpopular
continuousdistribution,thenormaldistribution.
GraphaNormalCurve
Manycontinuousrandomvariables,suchasIQscores,birthweightsofbabies,
orweightsofM&Ms,haverelativefrequencyhistogramsthathaveashapesim-
ilartoFigure3.Relativefrequencyhistogramsthathaveashapesimilarto
Figure3aresaidtohavetheshapeofa
normalcurve
Figure3
Definition
Acontinuousrandomvariableis
normallydistributed
orhas a
normalprobabilitydistribution
ifitsrelativefrequencyhistogram
oftherandomvariablehastheshapeofanormalcurve.
Figure4showsanormalcurve,demonstratingtherole and playin
drawingthecurve.LookbackatFigure5onpage113inSection3.1.Foranydis-
tribution,themoderepresentsthe“highpoint”ofthegraphofthedistribution.
Themedianrepresentsthepointwhere50%oftheareaunderthedistribution
istotheleftand50%oftheareaunderthedistributionistotheright.Themean
representsthebalancingpointofthegraphofthedistribution(seeFigure2on
page109inSection3.1).Forsymmetricdistributions,suchasthenormaldistri-
bution,the Becauseofthis,themean, isthe“high
point”ofthegraphofthedistribution.
Thepointsat and arethe
inflectionpoints
onthenor-
malcurve.The
inflectionpoints
arethepointsonthecurvewherethecurvature
Figure4
Inflection
point
Inflection
point
mean
=
median
=
mode.
#
smm
322
Chapter7TheNormalProbabilityDistribution
ofthegraphchanges.Totheleftof andtotherightof
thecurveisdrawnupward or .Inbetween and
thecurveisdrawndownward .*
Figure5showshowchangesinandchangethepositionorshapeofanormal
curve.InFigure5(a),twonormaldensitycurvesaredrawnwiththelocationofthe
inflectionpointslabeled.Onedensitycurvehas andtheotherhas
Wecanseethatincreasingthemeanfrom0to3causedthegraphto
shiftthreeunitstotheright.InFigure5(b),twonormaldensitycurvesaredrawn,
againwiththeinflectionpointslabeled.Onedensitycurvehas and
theotherhas Wecanseethatincreasingthestandarddeviationfrom
1to2causesthegraphtobecomeflatterandmorespreadout.
HistoricalNote
KarlPearsoncoinedthephrase
normalcurve
.Hedidnotdothisto
implythatadistributionthatisnot
normalis
abnormal.
Rather,Pearson
wantedtoavoidgivingthenameof
thedistributionapropername,such
asGaussian(asinCarlFriedrich
Gauss).
=
0,
s
=
1,
=
3,
s
=
1.
=
0,
s
=
1,
=
0,
s
=
2.
Figure5
m
$
0,
s
$
1
$
0,
s
$
3,
s
$
1
m
$
0,
s
#
1
0
3
4
#
4
0
2
4
6
(a)
(b)
NowWorkProblem25.
StatethePropertiesoftheNormalCurve
Thenormalprobabilitydensityfunctionsatisfiesalltherequirementsthatare
necessarytohaveaprobabilitydistribution.Welistthepropertiesofthenormal
densitycurvenext.
HistoricalNote
PropertiesoftheNormalDensityCurve
1.
Itissymmetricaboutitsmean,
2.
Because thehighestpointoccursat
3.
Ithasinflectionpointsat and
4.
Theareaunderthecurveis1.
5.
Theareaunderthecurvetotherightof equalstheareaunderthe
curvetotheleftof whichequals
6.
As
x
increases,withoutbound(getslargerandlarger),thegraphap-
proaches,butneverreaches,thehorizontalaxis.As
x
decreaseswithout
bound(getslargerandlargerinthenegativedirection),thegraphap-
proaches,butneverreaches,thehorizontalaxis.
7.
TheEmpiricalRule:Approximately68%oftheareaunderthenormal
curveisbetween and Approximately95%of
theareaunderthenormalcurveisbetween and
Approximately99.7%oftheareaunderthenormalcurve
isbetween and SeeFigure6.
AbrahamdeMoivrewasbornin
FranceonMay26,1667.Heisknown
asagreatcontributortotheareasof
probabilityandtrigonometry.In1685,
hemovedtoEngland.DeMoivrewas
electedafellowoftheRoyalSociety
in1697.Hewaspartofthe
commissiontosettlethedispute
betweenNewtonandLeibniz
regardingwhowasthediscovererof
calculus.Hepublished
TheDoctrine
ofChance
in1718.In1733,he
developedtheequationthatdescribes
thenormalcurve.Unfortunately,de
Moivrehadadifficulttimebeing
acceptedinEnglishsociety(perhaps
duetohisaccent)andwasableto
makeonlyameagerlivingtutoring
mathematics.Aninterestingpieceof
informationregardingdeMoivre;he
correctlypredictedthedayofhis
death,November27,1754.
mean
=
median
=
mode,
*Theverticalscaleonthegraph,whichindicatesdensity,ispurposelyomitted.Theverticalscale,
whileimportant,willnotplayaroleinanyofthecomputationsusingthiscurve.
[ Pobierz całość w formacie PDF ]
