15.3 - Exponential Examples | STAT 414

The number of miles that a particular car can run before its battery wears out is exponentially distributed with an average of 10,000 miles. Skiptomaincontent Breadcrumb Home 15 15.3 15.3-ExponentialExamples Example15-2 Section  StudentsarriveatalocalbarandrestaurantaccordingtoanapproximatePoissonprocessatameanrateof30studentsperhour.Whatistheprobabilitythatthebouncerhastowaitmorethan3minutestocardthenextstudent? Solution Ifwelet\(X\)equalthenumberofstudents,thenthePoissonmean\(\lambda\)is30studentsper60minutes,or\(\dfrac{1}{2}\)studentperminute!Now,ifwelet\(W\)denotethe(waiting)timebetweenstudents,wecanexpectthattherewouldbe,onaverage,\(\theta=\dfrac{1}{\lambda}=2\)minutesbetweenarrivingstudents.Because\(W\)is(assumedtobe)exponentiallydistributedwithmean\(\theta=2\),itsprobabilitydensityfunctionis: \(f(w)=\dfrac{1}{2}e^{-w/2}\) for\(w\ge0\).Now,wejustneedtofindtheareaunderthecurve,andgreaterthan3,tofindthedesiredprobability: Example15-3 Section  Thenumberofmilesthataparticularcarcanrunbeforeitsbatterywearsoutisexponentiallydistributedwithanaverageof10,000miles.Theownerofthecarneedstotakea5000-miletrip.Whatistheprobabilitythathewillbeabletocompletethetripwithouthavingtoreplacethecarbattery? Solution Atfirstglance,itmightseemthatavitalpieceofinformationismissing.Itseemsthatweshouldneedtoknowhowmanymilesthebatteryinquestionalreadyhasonitbeforewecananswerthequestion!Hmmm....ordowe?Well,let'slet\(X\)denotethenumberofmilesthatthecarcanrunbeforeitsbatterywearsout.Now,supposethefollowingistrue: \(P(X>x+y|X>x)=P(X>y)\) Ifitistrue,itwouldtellusthattheprobabilitythatthecarbatterywearsoutinmorethan\(y=5000\)milesdoesn'tmatterifthecarbatterywasalreadyrunningfor\(x=0\)milesor\(x=1000\)milesor\(x=15000\)miles.Now,wearegiventhat\(X\)isexponentiallydistributed.Itturnsoutthattheabovestatementistruefortheexponentialdistribution(youwillbeaskedtoproveitforhomework)!Itisforthisreasonthatwesaythattheexponentialdistributionis"memoryless." Itcanalsobeshown(doyouwanttoshowthatonetoo?)thatif\(X\)isexponentiallydistributedwithmean\(\theta\),then: \(P(X>k)=e^{-k/\theta}\) Therefore,theprobabilityinquestionissimply: \(P(X>5000)=e^{-5000/10000}=e^{-1/2}\approx0.604\) We'llleaveittothegentlemaninquestiontodecidewhetherthatprobabilityislargeenoughtogivehimcomfortthathewon'tbestrandedsomewherealongaremotedeserthighway! «Previous15.2-ExponentialProperties Next15.4-GammaDistributions» Lesson WelcometoSTAT414! Section1:IntroductiontoProbability Lesson1:TheBigPicture 1.1-SomeResearchQuestions 1.2-PopulationsandRandomSamples 1.3-SampleSpaces 1.4-Typesofdata 1.5-SummarizingQuantitativeDataGraphically Lesson2:PropertiesofProbability 2.1-WhyProbability? 2.2-Events 2.3-WhatisProbability(Informally)? 2.4-HowtoAssignProbabilitytoEvents 2.5-WhatisProbability(Formally)? 2.6-FiveTheorems 2.7-SomeExamples Lesson3:CountingTechniques 3.1-TheMultiplicationPrinciple 3.2-Permutations 3.3-Combinations 3.4-DistinguishablePermutations 3.5-MoreExamples Lesson4:ConditionalProbability 4.1-TheMotivation 4.2-WhatisConditionalProbability? 4.3-MultiplicationRule 4.4-MoreExamples Lesson5:IndependentEvents 5.1-TwoDefinitions 5.2-ThreeTheorems 5.3-MutualIndependence 5.4-AClosingExample Lesson6:Bayes'Theorem 6.1-AnExample 6.2-AGeneralization 6.3-AnotherExample 6.4-MoreExamples Section2:DiscreteDistributions Lesson7:DiscreteRandomVariables 7.1-DiscreteRandomVariables 7.2-ProbabilityMassFunctions 7.3-TheCumulativeDistributionFunction(CDF) 7.4-HypergeometricDistribution 7.5-MoreExamples Lesson8:MathematicalExpectation 8.1-ADefinition 8.2-PropertiesofExpectation 8.3-MeanofX 8.4-VarianceofX 8.5-SampleMeansandVariances Lesson9:MomentGeneratingFunctions 9.1-WhatisanMGF? 9.2-FindingMoments 9.3-FindingDistributions 9.4-MomentGeneratingFunctions Lesson10:TheBinomialDistribution 10.1-TheProbabilityMassFunction 10.2-IsXBinomial? 10.3-CumulativeBinomialProbabilities 10.4-EffectofnandponShape 10.5-TheMeanandVariance Lesson11:GeometricandNegativeBinomialDistributions 11.1-GeometricDistributions 11.2-KeyPropertiesofaGeometricRandomVariable 11.3-GeometricExamples 11.4-NegativeBinomialDistributions 11.5-KeyPropertiesofaNegativeBinomialRandomVariable 11.6-NegativeBinomialExamples Lesson12:ThePoissonDistribution 12.1-PoissonDistributions 12.2-FindingPoissonProbabilities 12.3-PoissonProperties 12.4-ApproximatingtheBinomialDistribution Section3:ContinuousDistributions Lesson13:ExploringContinuousData 13.1-Histograms 13.2-Stem-and-LeafPlots 13.3-OrderStatisticsandSamplePercentiles 13.4-BoxPlots 13.5-Shapesofdistributions Lesson14:ContinuousRandomVariables 14.1-ProbabilityDensityFunctions 14.2-CumulativeDistributionFunctions 14.3-FindingPercentiles 14.4-SpecialExpectations 14.5-Piece-wiseDistributionsandotherExamples 14.6-UniformDistributions 14.7-UniformProperties 14.8-UniformApplications Lesson15:Exponential,GammaandChi-SquareDistributions 15.1-ExponentialDistributions 15.2-ExponentialProperties 15.3-ExponentialExamples 15.4-GammaDistributions 15.5-TheGammaFunction 15.6-GammaProperties 15.7-AGammaExample 15.8-Chi-SquareDistributions 15.9-TheChi-SquareTable 15.10-TrickToAvoidIntegration Lesson16:NormalDistributions 16.1-TheDistributionandItsCharacteristics 16.2-FindingNormalProbabilities 16.3-UsingNormalProbabilitiestoFindX 16.4-NormalProperties 16.5-TheStandardNormalandTheChi-Square 16.6-SomeApplications Section4:BivariateDistributions Lesson17:DistributionsofTwoDiscreteRandomVariables 17.1-TwoDiscreteRandomVariables 17.2-ATriangularSupport 17.3-TheTrinomialDistribution Lesson18:TheCorrelationCoefficient 18.1-CovarianceofXandY 18.2-CorrelationCoefficientofXandY 18.3-UnderstandingRho 18.4-MoreonUnderstandingRho Lesson19:ConditionalDistributions 19.1-WhatisaConditionalDistribution? 19.2-Definitions 19.3-ConditionalMeansandVariances Lesson20:DistributionsofTwoContinuousRandomVariables 20.1-TwoContinuousRandomVariables 20.2-ConditionalDistributionsforContinuousRandomVariables Lesson21:BivariateNormalDistributions 21.1-ConditionalDistributionofYGivenX 21.2-JointP.D.F.ofXandY Section5:DistributionsofFunctionsofRandomVariables Lesson22:FunctionsofOneRandomVariable 22.1-DistributionFunctionTechnique 22.2-Change-of-VariableTechnique 22.3-Two-to-OneFunctions 22.4-SimulatingObservations Lesson23:TransformationsofTwoRandomVariables 23.1-Change-of-VariablesTechnique 23.2-BetaDistribution 23.3-FDistribution Lesson24:SeveralIndependentRandomVariables 24.1-SomeMotivation 24.2-ExpectationsofFunctionsofIndependentRandomVariables 24.3-MeanandVarianceofLinearCombinations 24.4-MeanandVarianceofSampleMean 24.5-MoreExamples Lesson25:TheMoment-GeneratingFunctionTechnique 25.1-UniquenessPropertyofM.G.F.s 25.2-M.G.F.sofLinearCombinations 25.3-SumsofChi-SquareRandomVariables Lesson26:RandomFunctionsAssociatedwithNormalDistributions 26.1-SumsofIndependentNormalRandomVariables 26.2-SamplingDistributionofSampleMean 26.3-SamplingDistributionofSampleVariance 26.4-Student'stDistribution Lesson27:TheCentralLimitTheorem 27.1-TheTheorem 27.2-ImplicationsinPractice 27.3-ApplicationsinPractice Lesson28:ApproximationsforDiscreteDistributions 28.1-NormalApproximationtoBinomial 28.2-NormalApproximationtoPoisson × Savechanges Close