AnnouncementsTopics:

IntheProbabilityandStatisticsmodule:-  Section10:TheBinomialDistribution-  Section13:ContinuousRandomVariables-  Section14:TheNormalDistribution

ToDo:-  WorkonAssignmentsandSuggestedPracticeProblemsassignedonthewebpageundertheSCHEDULE+HOMEWORKlink

BernoulliExperimentandBernoulliRandomVariable

ABernoulliexperimentisarandomexperimentwithonlytwopossibleoutcomes:successorno-success.Definition:Adiscreterandomvariablethattakesonthevalue1(“success”)withprobabilitypandthevalue0(“no-success”)withprobability1-piscalledaBernoullirandomvariable.

BernoulliExperimentandBernoulliRandomVariable

Example:ElephantPopulationwithImmigrationConsiderapopulationofelephantsptmodelledbyDefineaBernoulliexperimentanddeterminetheprobabilitymassfunctionforthecorrespondingBernoullirandomvariable.

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pt+1 = pt + It where It =10 with a 90% chance0 with a 10% chance" # $

TheBinomialDistribution

LetNcountthenumberofsuccessesinnrepetitionsofthesameBernoulliexperiment,whereoutcomesareindependentandpistheprobabilityofsuccessinasingleexperiment.ThenNisabinomiallydistributedrandomvariableandwewrite

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N ~ B(n, p)

TheBinomialDistribution

Definethebinomialprobabilitydistributionbywhereistheprobabilityofexactlyksuccessesinnrepetitionsofthesameexperiment,wherepistheprobabilityofsuccessinasingleexperiment.

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b(k, n; p) = P(N = k)

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b(k, n; p)

TheBinomialDistribution

Example:ElephantPopulationwithImmigrationConsiderthepopulationofelephantsptmodelledbywheret=0,1,2,…ismeasuredinyears.LetNcountthenumberoftimesimmigrationoccursoverthenext3years.DeterminetheprobabilitymassfunctionforN.

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pt+1 = pt + It where It =10 with a 90% chance0 with a 10% chance" # $

TheBinomialDistribution

Theprobabilityofksuccessesinnexperimentsis(numberofwaysofobtainingksuccessesinnexperiments)*(probabilityofsuccess)k*(probabilityofno-success)n-ki.e.,

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b(k, n; p) = C(n,k)pk (1− p)n−k

Counting101

Supposewehaveaselectionof10books.1.  Howmanydifferentorderings(permutations)canyoureadthemin?

2.  Howmanydifferentorderingscanyouread3in?

3.  Supposeyouwanttoreadthreebooksbutyoudon’tcareabouttheorderinwhichyoureadthem.Howcanyouchoose3booksfrom10?

Counting101

Now,replace“10books”by“nexperiments”and“choose3books”by“choosekexperimentsinwhichthereisasuccess”.ThenumbersofwayswecanhaveksuccessesinnrepetitionsoftheexperimentisnCk.So,C(n,k)=nCk.

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TheProbabilityDistributionoftheBinomialVariable

Theorem:TheprobabilitydistributionofthebinomialvariableNisgivenbywhereNcountsthenumberofsuccessesinnindependentrepetitionsofthesameBernoulliexperimentandpistheprobabilityofsuccess.

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P(N = k) = b(k, n; p) =nk"

# $ %

& ' pk (1− p)n−k

TheProbabilityDistributionoftheBinomialVariable

Example:CoinTossWhatistheprobabilityofexactly7tailsin10tosses?

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TheProbabilityDistributionoftheBinomialVariable

Example:ElephantPopulationwithImmigrationConsiderapopulationofelephantsptmodelledbySupposethatinitiallythereare80elephants.Whatistheprobabilitythattherewillbemorethan300elephantsafter25years?

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pt+1 = pt + It where It =10 with a 90% chance0 with a 10% chance" # $

TheMeanandVarianceoftheBinomialDistribution

MeanandVarianceoftheBinomialRandomVariableN:

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E(N ) = npVar(N ) = np(1− p)

TheMeanandVarianceoftheBinomialDistribution

Example:ElephantPopulationwithImmigrationConsiderapopulationofelephantsptmodelledbySupposethatinitiallythereare80elephants.Whatistheexpectedvalueofthepopulationafter25years?Whatisthestandarddeviation?

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pt+1 = pt + It where It =10 with a 90% chance0 with a 10% chance" # $

ContinuousRandomVariables

Definition:Arandomvariablethattakesonacontinuumofvaluesiscalledacontinuousrandomvariable.

ContinuousRandomVariables

Example:DistributionsofLengthsofBoaConstrictorsTheboaconstrictorisalargespeciesofsnakethatcangrowtoanywherebetween1mand4minlength.LetLbethecontinuousrandomvariablethatmeasuresthelengthofasnake.

L : S→ [1, 4]

ContinuousRandomVariables

Thelengthsof500boasarerecordedbelow:Note:relativefrequency=frequency/500=probability

ContinuousRandomVariablesHistogramforProbabilityMass:Theprobabilitythatarandomlyselectedboaisbetween2.5mand3minlengthistheheightoftherectangleover[2.5,3),i.e.,0.36.

ContinuousRandomVariables

Todrawahistogramrepresentingprobabilitydensity,were-labeltheverticalaxissothattheprobabilitythatLbelongstoanintervalistheareaoftherectangleabovethatinterval.

ContinuousRandomVariables

Note: probabilitymass lengthofinterval

=probabilitydensity

ContinuousRandomVariables

Forexample,considertheinterval[2.5,3).TheprobabilitythatLfallsinthisrangeis0.36.Now,wewantthisvaluetobetheareaoftherectangleover[2.5,3),soprobabilitydensity(height)=0.36/(3-2.5)=0.72

ContinuousRandomVariables

HistogramforProbabilityDensity:Theprobabilitythatarandomlyselectedboaisbetween2.5mand3minlengthistheareaoftherectangleabove[2.5,3),i.e.0.36.

ContinuousRandomVariablesTogetamorepreciseprobabilitymass(ordensity)function,wedivide[1,4]intosmallersubintervals:.

ContinuousRandomVariables

Aswecontinuetoincreasethenumberofsubintervals,weobtainamoreandmorerefinedhistogram..

ContinuousRandomVariablesRiemannSum:Theprobabilitythatarandomlychosenboaisbetween1.75mand2minlengthisthesumoftheareasoftherectanglesovertheinterval[1.75,2).

ContinuousRandomVariables

Toobtaintheprobabilitydensityfunction,weletthelengthoftheintervalsapproach0andthenumberofrectanglesapproach∞..

ProbabilityDensityFunctions

Definition:DefiningPropertiesofaPDFAssumethattheintervalIrepresentstherangeofacontinuousrandomvariableX.Afunctionf(x)canbeaprobabilitydensityfunctionif(1)(2)

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f (x) ≥ 0 for all x ∈ I.

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f (x)dx =1.I∫

ProbabilityDensityFunctions

Example:Showthatcouldbeaprobabilitydensityfunctionforsomecontinuousrandomvariableon[0,∞).

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f (x) =2

π (1+ x 2)

CalculatingProbabilitiesForacontinuousrandomvariable,wecalculatetheprobabilitythatarandomvariablebelongstoanintervalofrealnumbers.TheprobabilitythatanoutcomeXisbetweenaandbistheareaunderthegraphoff(x)on[a,b]:

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P(a ≤ X ≤ b) = f (x)dxa

b

∫

CalculatingProbabilities

Theprobabilitythatanoutcomeisequaltoaparticularvalueiszero.Forthisreason,includingorexcludingtheendpointsofanintervaldoesnotaffecttheprobability,i.e.,

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P(a ≤ X ≤ a) = f (x)dx = 0a

a

∫

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P(a ≤ X ≤ b) = P(a ≤ X < b) = P(a < X ≤ b) = P(a < X < b)

CalculatingProbabilitiesExample#32:Thedistancebetweenaseedandtheplantitcamefromismodelledbythedensityfunctionwherexrepresentsthedistance(inmetres),Whatistheprobabilitythataseedwillbefoundfartherthan5mfromtheplant?

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f (x) =2

π (1+ x 2)

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x ∈ [0,∞).

CumulativeDistributionFunction

Definition:Supposethatf(x)isaprobabilitydensityfunctiondefinedonaninterval[a,b].ThefunctionF(x)definedbyforallxin[a,b]iscalledacumulativedistributionfunctionoff(x). €

F(x) = P(X ≤ x) = f (t)dta

x

∫

CumulativeDistributionFunctionExample#30(modified):Supposethatthelifetimeofaninsectisgivenbytheprobabilitydensityfunctionwheretismeasuredindays,(a)Determinethecorrespondingcumulativedistributionfunction,F(t).(b)Findtheprobabilitythattheinsectwilllivebetween5-7days.

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f (t) = 0.2e−0.2t

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t ∈ [0,∞).

CumulativeDistributionFunctionExample#30(modified):

PROBABILITY�DENSITY�FUNCTION�F�T� � � ���������CUMULATIVE�DISTRIBUTION�FUNCTION�&�T

CumulativeDistributionFunction

PropertiesoftheCDF:Assumethatfisaprobabilitydensityfunction,definedandcontinuousonaninterval[a,b].Theleftendacouldbearealnumberornegativeinfinity;therightendbcouldbearealnumberorinfinity.DenotebyFtheassociatedcumulativedistributionfunction.Then(1)(2)iscontinuousandnon-decreasing.(3)

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0 ≤ F(x) ≤1 for all x ∈ [a,b].

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F(x)

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limx→a

F(x) = 0 and limx→b

F(x) =1.

TheMeanandtheVarianceDefinition:LetXbeacontinuousrandomvariablewithprobabilitydensityfunctionf(x),definedonaninterval[a,b].Themean(ortheexpectedvalue)ofXisgivenbyThevarianceofXis €

µ = E(X) = x f (x)dxa

b

∫

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var(X) = E (X −µ)2[ ] = (x −µ)2 f (x)dxa

b

∫

TheMeanandtheVarianceExample#24:ConsiderthecontinuousrandomvariableXgivenbytheprobabilitydensityfunctionFindtheprobabilitythatthevaluesofXareatleastonestandarddeviationabovethemean.

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f (x) = 0.3+ 0.2x for 0 ≤ x ≤ 2.

TheNormalDistribution

Thenormaldistributionisthemostimportantcontinuousdistributionasitcanbeusedtomodelmanyphenomenainavarietyoffields.Manymeasurementsforlargesamplesizesaresaidtobe‘normallydistributed’.Forexample,heightsoftrees,IQscores,anddurationofpregnancyareallnormallydistributedmeasurements.

TheNormalDistribution

Definition:AcontinuousrandomvariableXhasanormaldistribution(orisdistributednormally)withmeanandvariance,denotedby,ifitsprobabilitydensityfunctioniswhere

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f (x) =1

σ 2πe−(x−µ )2

2σ 2

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µ

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σ 2

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X ~ N(µ,σ 2)

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x ∈ (−∞,∞).

TheNormalDistribution

Thegraphoftheprobabilitydensityfunctionofthenormaldistribution(alsoknownastheGaussiandistribution)isabell-shapedcurve.

PropertiesoftheNormalDistributionDensityFunction

Theorem:Theprobabilitydensityfunctionf(x)ofthenormaldistributionsatisfiesthefollowingproperties:(a) f(x)issymmetricwithrespecttotheverticalline(b)  f(x)isincreasingforanddecreasingforIthasalocal(alsoglobal)maximumvalueat(c)Theinflectionpointsoff(x)are(d)

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limx→−∞

f (x) = limx→∞

f (x) = 0

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x = µ.

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x < µ

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x > µ.

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1 σ 2π

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x = µ ±σ .

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x = µ.

CalculatingProbabilities

IfXisanormallydistributedcontinuousrandomvariablewithmeanandvariancethen

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P(a ≤ X ≤ b) = f (x)dx =a

b

∫ 1σ 2π

e−(x−µ )2

2σ 2 dxa

b

∫€

µ

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σ 2,

CalculatingProbabilities

Thisintegralcannotbeevaluatedwithoutestimationtechniques,suchasusingaTaylorpolynomialtoapproximatef(x).Toevaluatethisintegral,wereduceageneralnormaldistributiontoaspecialnormaldistribution,calledthestandardnormaldistribution,andthenusetablesofestimatedvalues.

StandardNormalDistribution

Definition:Thestandardnormaldistributionisthenormaldistributionwithmean0andvariance1;insymbols,itisN(0,1).Itsprobabilitydensityfunctionisgivenbyforall €

f (x) =12π

e−x 2

2

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x ∈ (−∞,∞).

StandardNormalDistribution

WeusethesymbolZtodenotethecontinuousrandomvariablethathasthestandardnormaldistribution;i.e.,

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Z ~ N(0,1).

StandardNormalDistribution

ThecumulativedistributionfunctionofZisgivenby

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F(z) = f (x)dx =−∞

z

∫ 12π

e−x 2

2 dx−∞

z

∫

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F(−z) =1− F(z)

TheNormalandtheStandardNormalDistributions

Theorem:AssumethatTherandomvariableZ=hasthestandardnormaldistribution,i.e.,Sothen

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X ~ N(µ,σ 2).

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Z ~ N(0,1).

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Z = (X −µ) σ

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P(a ≤ X ≤ b) = P(a −µσ

≤ Z ≤ b −µσ).

TheNormalandtheStandardNormalDistributions

Inwords,theareaunderthenormaldistributiondensityfunctionbetweenaandbisequaltotheareaunderthestandardnormaldensityfunctionbetweenand

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(a −µ) /σ

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(b −µ) /σ .

TheNormalandtheStandardNormalDistributions

Example#10:Example#30:

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Let X ~ N(−2,4); find P(−3 ≤ X ≤1).

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Let X ~ N(2,144); find a value of x that satisfies P(X > x) = 0.3.

Application

Example:Intelligencequotient(IQ)scoresaredistributednormallywithmean100andstandarddeviation15.(a)  WhatpercentageofthepopulationhasanIQscorebetween85and115?

(b)  WhatpercentageofthepopulationhasanIQabove140?

(c)  WhatIQscoredo90%ofpeoplefallunder?

68-95-99.7Rule

IfXisacontinuousrandomvariabledistributednormallywithmeanandstandarddeviation,then

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P(µ −σ ≤ X ≤ µ +σ) = 0.683P(µ − 2σ ≤ X ≤ µ + 2σ) = 0.955P(µ − 3σ ≤ X ≤ µ + 3σ) = 0.997€

µ

€

σ

68-95-99.7Rule

Inwords,foranormallydistributedrandomvariable:68.3%ofthevaluesfallwithinonestandarddeviationofthemean.95.5%ofthevaluesfallwithintwostandarddeviationsofthemean.99.7%ofthevaluesfallwithinthreestandarddeviationsofthemean.

Application

Example14.7:TheLengthsofPregnanciesThelengthsofhumanpregnancies(measuredindaysfromconceptiontobirth)canbeapproximatedbythenormaldistributionwithameanof266daysandastandarddeviationof16days.

Application

Example14.7:TheLengthsofPregnanciesThus,about68%ofpregnancieslastbetween266-16=250daysand266+16=282days.About95.5%ofpregnancieslastbetween266-2x16=234daysand266+2x16=298days,andabout99.7%ofpregnancieslastbetween266-3x16=218daysand266+3x16=314days.