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Lecture 7

# Lecture 7

9 Pages
134 Views

Department
Quantitative Methods
Course Code
QMS 102
Professor
Clare Chua

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10/19/2010
1
Probability
Topicsî€ƒcoveredî€ƒtoday:
1. Basicî€ƒProbabilityî€ƒtheoryî€ƒ(Chapterî€ƒ4î€ƒ)
2. Discreteî€ƒProbabilityî€ƒDistributionî€ƒ
(Chapterî€ƒ5,î€ƒsection5.1)
Typesî€ƒofî€ƒProbability
Discrete Continuous
Numerical
(Random
variable)
Categorical
Data
Chapters 1,2,3
Binomial
Hypergeometric
Poisson
Probability
Distribution
Normal
Distrbution
Uniform
Distribution
Exponential
DIstribution
Probability
Distribution
Chapters
4,5, 6
Probability
to cope
with uncertainty
Definitions
â€¢Probability: Aî€ƒmeasureî€ƒofî€ƒtheî€ƒlikelihoodî€ƒthatî€ƒanî€ƒeventî€ƒ
willî€ƒhappen.î€ƒî€ƒ(Youî€ƒcanî€ƒalsoî€ƒreferî€ƒtoî€ƒtheî€ƒprobabilityî€ƒofî€ƒanî€ƒeventî€ƒ
whetherî€ƒorî€ƒnotî€ƒtheî€ƒeventî€ƒoccurred.)
â€¢Experiment:Somethingî€ƒdoneî€ƒtoî€ƒobtainî€ƒdata.î€ƒî€ƒItî€ƒmayî€ƒ
involveî€ƒaî€ƒphysicalî€ƒactionî€ƒtoî€ƒmakeî€ƒsomethingî€ƒtakeî€ƒplaceî€ƒorî€ƒmayî€ƒ
j
ustî€ƒinvolveî€ƒobservin
g
î€ƒsomethin
g
î€ƒha
pp
en.
jg g pp
â€¢Outcome: Theî€ƒresultî€ƒofî€ƒoneî€ƒtrialî€ƒofî€ƒtheî€ƒexperiment.
â€¢Sampleî€ƒSpace: Theî€ƒlistî€ƒofî€ƒallî€ƒpossibleî€ƒoutcomes.
â€¢Event:Aî€ƒcollectionî€ƒofî€ƒoneî€ƒorî€ƒmoreî€ƒoutcomes.
Exampleî€ƒ1:î€ƒ
a)î€ƒonce.Experiment:î€ƒTossî€ƒcoinî€ƒî€ƒoneî€ƒtime
Sampleî€ƒSpace:î€ƒî€ƒ{Hî€ƒî€ƒT}
b)î€ƒtwice.î€ƒ(orî€ƒtossî€ƒ2î€ƒcoins) Experiment:î€ƒî€ƒTossî€ƒcoinî€ƒtwice.
Sam
p
leî€ƒS
p
ace:î€ƒî€ƒ
{
HHî€ƒî€ƒî€ƒHTî€ƒî€ƒî€ƒTHî€ƒî€ƒî€ƒTT
}
p p { }
c)î€ƒthreeî€ƒtimes.î€ƒ(orî€ƒtossî€ƒ3î€ƒcoins)î€ƒExperiment:î€ƒî€ƒTossî€ƒcoinî€ƒ3î€ƒtimes
Sampleî€ƒSpace:î€ƒî€ƒ{HHHî€ƒî€ƒHHTî€ƒî€ƒHTHî€ƒî€ƒTHHî€ƒî€ƒTTHî€ƒî€ƒTHTî€ƒî€ƒHTTî€ƒî€ƒTTT}
Exampleî€ƒ2:
Rollî€ƒaî€ƒdie: (Interestedî€ƒinî€ƒtheî€ƒresultî€ƒofî€ƒeachî€ƒdie.)
a)î€ƒonce.î€ƒExperiment:î€ƒRollî€ƒdieî€ƒonce
Sampleî€ƒSpace:î€ƒî€ƒ{1î€ƒî€ƒî€ƒ2î€ƒî€ƒî€ƒ3î€ƒî€ƒî€ƒ4î€ƒî€ƒî€ƒ5î€ƒî€ƒî€ƒ6}
Event:î€ƒî€ƒe.g.î€ƒGetî€ƒanî€ƒevenî€ƒnumberî€ƒr{2î€ƒî€ƒ4î€ƒî€ƒ6}
b)î€ƒtwice.î€ƒî€ƒ(orî€ƒrollî€ƒ2î€ƒdice)î€ƒExperiment:î€ƒî€ƒRollî€ƒdieî€ƒtwice.î€ƒ
Sampleî€ƒSpace:î€ƒî€ƒ{1r1î€ƒî€ƒ1r2î€ƒî€ƒ1r3î€ƒî€ƒ1r4î€ƒî€ƒ1r5î€ƒî€ƒ1r6
2r1î€ƒî€ƒ2r2î€ƒî€ƒ2r3î€ƒî€ƒ2r4î€ƒî€ƒ2r5î€ƒî€ƒ2r6
3r1î€ƒî€ƒ3r2î€ƒî€ƒ3r3î€ƒî€ƒ3r4î€ƒî€ƒ3r5î€ƒî€ƒ3r6
4r1î€ƒî€ƒ4r2î€ƒî€ƒ4r3î€ƒî€ƒ4r4î€ƒî€ƒ4r5î€ƒî€ƒ4r6
5r1î€ƒî€ƒ5r2î€ƒî€ƒ5r3î€ƒî€ƒ5r4î€ƒî€ƒ5r5î€ƒî€ƒ5r6
6r1î€ƒî€ƒ6r2î€ƒî€ƒ6r3î€ƒî€ƒ6r4î€ƒî€ƒ6r5î€ƒî€ƒ6r6}
Event:î€ƒî€ƒe.g.î€ƒGetî€ƒaî€ƒtotalî€ƒofî€ƒ5î€ƒr{1r4î€ƒî€ƒ2r3î€ƒî€ƒ3r2î€ƒî€ƒ4r1}
Probability
a)î€ƒî€ƒSymbol:î€ƒî€ƒî€ƒP(event)î€ƒ=î€ƒî€ƒ#
Chance isî€ƒexpressedî€ƒasî€ƒaî€ƒpercent.
c)î€ƒî€ƒIfî€ƒP(event)î€ƒ=î€ƒ0î€ƒî€ƒthenî€ƒitî€ƒisî€ƒnotî€ƒpossible forî€ƒtheî€ƒeventî€ƒtoî€ƒoccur.
Ifî€ƒP(event)î€ƒ=î€ƒ1î€ƒî€ƒthenî€ƒtheî€ƒeventî€ƒisî€ƒaî€ƒcertaintyi.e.î€ƒguaranteedî€ƒtoî€ƒoccur.
d)î€ƒTheî€ƒsumî€ƒofî€ƒallî€ƒtheî€ƒprobabilitiesî€ƒinî€ƒtheî€ƒsampleî€ƒspaceî€ƒisî€ƒ1
www.notesolution.com
10/19/2010
2
Probability
Not possibleCertainty
0.0 0.10.2 0.30.4 0.50.6 0.70.8 0.91.0
Prob. our sun will
disappear this
year
Prob. the
Leafs will win
the cup
Prob. of
getting a
coin toss
Prob. of an
increase in
taxes this
year
Prob. of snow in
Toronto this
winter
1. Allî€ƒprobabilitiesî€ƒareî€ƒbetweenî€ƒ0î€ƒandî€ƒ1î€ƒinclusive
2. Theî€ƒprobabilityî€ƒofî€ƒanî€ƒeventî€ƒwhichî€ƒcannotî€ƒoccurî€ƒisî€ƒ0.
3. Theî€ƒprobabilityî€ƒofî€ƒanî€ƒeventî€ƒwhichî€ƒoccurî€ƒwithî€ƒ
certaintyî€ƒisî€ƒ1.
4. Theî€ƒprobabilityî€ƒofî€ƒanî€ƒeventî€ƒnotî€ƒoccurringî€ƒisî€ƒoneî€ƒ
minusî€ƒtheî€ƒprobabilityî€ƒofî€ƒitî€ƒoccurring.
P(E')î€ƒ=î€ƒ1î€ƒrP(E)î€ƒ
5.î€ƒYouî€ƒcanî€ƒalsoî€ƒrepresentî€ƒprobabilityî€ƒasî€ƒaî€ƒdecimalî€ƒorî€ƒasî€ƒaî€ƒ
percent.
Approachesî€ƒtoî€ƒProbability
A) Subjective
B) Experimentalî€ƒ(orî€ƒrelativeî€ƒfrequency)
C) Theoreticalî€ƒ(orî€ƒClassical)
A)î€ƒî€ƒSubjective:
Personalî€ƒjudgmentî€ƒandî€ƒexpressesî€ƒdegreeî€ƒofî€ƒ
belief.
l
Examp
l
e:
P(Snowî€ƒtoday)î€ƒ=î€ƒ0.01î€ƒî€ƒî€ƒBasedî€ƒonî€ƒstudyî€ƒofî€ƒ
weatherî€ƒconditions.î€ƒî€ƒNotî€ƒcalculatedî€ƒthroughî€ƒ
theî€ƒuseî€ƒofî€ƒprobabilityî€ƒformula.
B)î€ƒî€ƒExperimentalî€ƒ(orî€ƒrelativeî€ƒfrequency):
Definition:î€ƒTheî€ƒchancesî€ƒofî€ƒsomeî€ƒeventî€ƒoccurredî€ƒandî€ƒweî€ƒeitherî€ƒconductî€ƒrepeatedî€ƒ
testingî€ƒorî€ƒobserveî€ƒî€ƒevent.î€ƒTheî€ƒprobabilityî€ƒisî€ƒtheî€ƒratioî€ƒofî€ƒtheî€ƒnumberî€ƒofî€ƒtimesî€ƒanî€ƒ
eventî€ƒoccurredî€ƒtoî€ƒtheî€ƒnumberî€ƒofî€ƒtimesî€ƒtested.
Exampleî€ƒ1:î€ƒî€ƒToss î€ƒaî€ƒthumbtack.î€ƒî€ƒFindî€ƒtheî€ƒprobabilityî€ƒthatî€ƒtheî€ƒthumbtackî€ƒwillî€ƒlandî€ƒonî€ƒitsî€ƒ
Theî€ƒonlyî€ƒwayî€ƒtoî€ƒdetermineî€ƒthisî€ƒprobabilityî€ƒisî€ƒtoî€ƒtossî€ƒtheî€ƒthumbtackî€ƒandî€ƒcountî€ƒtheî€ƒ
numberî€ƒofî€ƒtimesî€ƒitî€ƒlandsî€ƒwithî€ƒitsî€ƒpointî€ƒstickingî€ƒup.î€ƒî€ƒ
infiniteî€ƒnumberî€ƒofî€ƒtimes.î€ƒî€ƒ
Dependingî€ƒonî€ƒtheî€ƒaccuracyî€ƒrequired,î€ƒweî€ƒcanî€ƒdetermineî€ƒtheî€ƒnumberî€ƒofî€ƒtimesî€ƒtheî€ƒ
experimentî€ƒmustî€ƒbeî€ƒrepeated.
Exampleî€ƒ2: Inî€ƒtheî€ƒpastî€ƒfewî€ƒmonths,î€ƒ39%î€ƒofî€ƒtheî€ƒtrucksî€ƒthatî€ƒhaveî€ƒbeenî€ƒinspectedî€ƒatî€ƒaî€ƒ
particularî€ƒinspectionî€ƒstationî€ƒhaveî€ƒfailedî€ƒtheî€ƒsafetyî€ƒinspection.î€ƒî€ƒWhatî€ƒisî€ƒtheî€ƒ
probabilityî€ƒthatî€ƒtheî€ƒnextî€ƒtruckî€ƒinspectedî€ƒwillî€ƒfailî€ƒtheî€ƒsafetyî€ƒinspection?
Theoreticalî€ƒ(orî€ƒClassical)
Definitionî€ƒ:
Probabilityî€ƒisî€ƒaî€ƒlikelihoodî€ƒthatî€ƒanî€ƒeventî€ƒwillî€ƒhappen.
Theî€ƒtheoreticalî€ƒprobabilityî€ƒofî€ƒanî€ƒeventî€ƒcanî€ƒbeî€ƒcalculatedî€ƒas:
Ifî€ƒallî€ƒtheî€ƒoutcomesî€ƒinî€ƒtheî€ƒsampleî€ƒspaceî€ƒareî€ƒequallyî€ƒlikely,î€ƒthen
space
sample
the
in
outcomes
of
event thein outcomes ofnumber
)( eventP
Example:
Ifî€ƒyouî€ƒtossî€ƒaî€ƒfairî€ƒcoin,î€ƒwhatî€ƒisî€ƒtheî€ƒprobabilityî€ƒthatî€ƒaî€ƒtailî€ƒwillî€ƒshowî€ƒup?
Solution:
Soî€ƒtheî€ƒoptionsî€ƒforî€ƒtossingî€ƒaî€ƒtailî€ƒareî€ƒ1î€ƒoutî€ƒofî€ƒ2.
space
sample
the
in
outcomes
of
Continuedâ€¦Theoreticalî€ƒ(orî€ƒClassical)
Exampleî€ƒ:
Aî€ƒbagî€ƒcontainsî€ƒ100î€ƒmarbles.î€ƒThereî€ƒareî€ƒ30î€ƒgreenî€ƒ
marblesî€ƒandî€ƒ70î€ƒyellowî€ƒmarbles.î€ƒFindî€ƒtheî€ƒ
probabilityî€ƒofî€ƒpickingî€ƒaî€ƒgreenî€ƒmarble.
30
bag
the
in
marbles
red
of
Number
3.0
100
30
)( bagtheinmarblesofNumberTotal
bag
the
in
marbles
red
of
Number
marblegreenp
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10/19/2010
3
Discreteî€ƒProbabilityî€ƒDistribution
Probabilityî€ƒDistributionî€ƒForî€ƒAî€ƒDiscreteî€ƒ
Randomî€ƒVariable
Aî€ƒprobabilityî€ƒdistributionî€ƒforî€ƒaî€ƒdiscreteî€ƒrandomî€ƒvariableî€ƒisî€ƒaî€ƒ
mutuallyî€ƒexclusiveî€ƒlistingî€ƒofî€ƒallî€ƒtheî€ƒpossibleî€ƒnumericalî€ƒ
outcomesî€ƒalongî€ƒwithî€ƒtheî€ƒprobabilityî€ƒofî€ƒoccurrenceî€ƒofî€ƒeachî€ƒ
outcome.î€ƒProbabilityî€ƒofî€ƒ
Distribution
X=Numberî€ƒ
ofî€ƒtails P(X)
01/8
13/8
23/8
31/8
Probability
Probabilityî€ƒofî€ƒ
occurrence
Probabilityî€ƒofî€ƒ
Distribution
p(E) =î€ƒî€ƒX
T
P(X)
1/8
X=Numberî€ƒ
ofî€ƒtails
T
Xî€ƒ=î€ƒ#î€ƒofî€ƒwaysî€ƒinî€ƒwhichî€ƒ
theî€ƒeventî€ƒoccurs
Tî€ƒ=î€ƒtotalî€ƒ#î€ƒofî€ƒoutcomesî€ƒ
Example:î€ƒTossî€ƒaî€ƒcoinî€ƒandî€ƒguess
0
1
2
3
1/8
3/8
1/8
3/8
Tossî€ƒaî€ƒfairî€ƒcoinî€ƒ3î€ƒtimesî€ƒ
Eî€ƒ=î€ƒEvent
Objective
1. Identifyî€ƒtheî€ƒProbabilityî€ƒofî€ƒDistribution
2. Com
p
uteî€ƒtheî€ƒMeanî€ƒandî€ƒStandardî€ƒ
p
Deviation
Randomî€ƒVariable
Whatî€ƒisî€ƒRandomî€ƒVariable?
Whatî€ƒisî€ƒRandomî€ƒVariable?
Definition:
Aî€ƒrandomî€ƒvariableî€ƒisî€ƒaî€ƒvariableî€ƒthatî€ƒtakesî€ƒonî€ƒ
differentî€ƒvaluesî€ƒasî€ƒaî€ƒresultî€ƒofî€ƒtheî€ƒoutcomesî€ƒofî€ƒ
aî€ƒrandomî€ƒex
p
eriment.
p
Notation:î€ƒX=î€ƒRandomî€ƒVariable
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Description
10192010 TypesofProbability Data Chapters 1,2,3 Numerical Categorical (Random variable) Probability Discrete Continuous Distribution Distribution Topicscoveredtoday: 1. BasicProbabilitytheory(Chapter4) Chapters Binomial Normal Distrbution 2. DiscreteProbabilityDistribution 4,5, 6 (Chapter5,section5.1) Probability Hypergeometric Distribution to cope with uncertainty Poisson Exponential DIstribution Definitions Example1: Probability: Ameasureofthelikelihoodthatanevent Tossacoin:(Interestedinresults,thatis,headstails.) willhappen.(Youcanalsorefertotheprobabilityofanevent a)once. Experiment:Toss coinonetime thatmayhavealreadyoccurred,butonlyifyoudontknow whetherornottheeventoccurred.) SampleSpace:{HT} Event:e.g.GetaheadH {H} Experiment: Somethingdonetoobtaindata.Itmay involveaphysicalactiontomakesomethingtakeplaceormay b)twice.(ortoss2coins) Experiment:Tosscointwice. jjustinvolveobservigsomething ghappen. Sam plepace:{{HHHTTHTT}} Event:e.g.GetatleastoneheadH {HHHTTH} Outcome: Theresultofonetrialoftheexperiment. SampleSpace: Thelistofallpossibleoutcomes. c)threetimes.(ortoss3coins)Experiment:Tosscoin3times SampleSpace:{HHHHHTHTHTHHTTHTHTHTTTTT} Event: Acollectionofoneormoreoutcomes. Event:e.g.GetoneheadH {HTTTHTTTH} Example2: Probability Rolladie: (Interestedintheresultofeachdie.) a)Symbol:P(event)=# a)once.Experiment:Rolldieonce b)Answer:Decimalorfractionbetween0and1.(Decimalpreferred) Chance isexpressedasapercent. SampleSpace:{123456} Event:e.g.GetanevennumberH {246} e.g.Tossacoin.P(head)=0.5 Thereisa50%chanceofgettingahead. b)twice.(orroll2dice)Experiment:Rolldietwice. SampleSpace:{1H11H21H31H4H51H6 c)IfP(event)=0thenitisnotpossibleventtooccur. 2H12H22H32H42H52H6 IfP(event)=1thentheeventisacertainty i.e.guaranteedtooccur. 3H13H23H33H43H53H6 d)Thesumofalltheprobabilitiesinthesamplespaceis1 4H14H24H34H44H54H6 5H15H25H35H45H55H6 6H16H2636H46H56H6} Event:e.g.Getatotalof5H {1H42H33H24H1} 1 www.notesolution.com
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