Exponential distribution | Properties, proofs, exercises - StatLect

Expected value StatLect Index>Probabilitydistributions Exponentialdistribution byMarcoTaboga,PhD Theexponentialdistributionisacontinuousprobabilitydistributionusedto modelthetimeelapsedbeforeagiveneventoccurs. Sometimesitisalsocallednegativeexponentialdistribution. Tableofcontents HowthedistributionisusedWaitingtimeDefinitionTherateparameteranditsinterpretationExpectedvalueVarianceMomentgeneratingfunctionCharacteristicfunctionDistributionfunctionMoredetailsMemorylesspropertyThesumofexponentialrandomvariablesisaGammarandomvariableRelationtothePoissondistributionDiscretecounterpartDensityplotSolvedexercisesExercise1Exercise2Exercise3 Howthedistributionisused Theexponentialdistributionisoftenusedtoanswerinprobabilisticterms questionssuchas: Howmuchtimewillelapsebeforeanearthquakeoccursinagivenregion? Howlongdoweneedtowaituntilacustomerentersourshop? Howlongwillittakebeforeacallcenterreceivesthenextphonecall? Howlongwillapieceofmachineryworkwithoutbreakingdown? Allthesequestionsconcernthetimeweneedtowaitbeforeagivenevent occurs. Ifthiswaitingtimeisunknown,itisoftenappropriatetothinkofitasa randomvariablehavinganexponentialdistribution. Waitingtime Awaitingtime hasanexponentialdistributioniftheprobabilitythattheeventoccurs duringacertaintimeintervalisproportionaltothelengthofthattime interval. Moreprecisely, hasanexponentialdistributioniftheconditional probabilityis approximatelyproportionaltothelength ofthetimeintervalcomprisedbetweenthetimes and , foranytimeinstant . Inseveralpracticalsituationsthispropertyisrealistic.Thisisthereason whytheexponentialdistributioncanbeusedtomodelwaitingtimes. Definition Theexponentialdistributionischaracterizedasfollows. Definition Let beacontinuous randomvariable.Letits supportbetheset ofpositivereal numbers:Let . Wesaythat hasanexponentialdistributionwithparameter ifandonlyifits probabilitydensity function isThe parameter iscalledrateparameter. Arandomvariablehavinganexponentialdistributionisalsocalledan exponentialrandomvariable. Thefollowingisaproofthat isalegitimateprobabilitydensityfunction. Proof Non-negativityisobvious.Weneedtoprove thattheintegralof over equals . Thisisprovedas follows: Tobetterunderstandtheexponentialdistribution,youcanhavealookatits densityplots. Therateparameteranditsinterpretation Wehavementionedthattheprobabilitythattheeventoccursbetweentwodates and isproportionalto (conditionalontheinformationthatithasnotoccurredbefore ). Therateparameter istheconstantof proportionality:where isaninfinitesimalofhigherorderthan (i.e.afunctionof thatgoestozeromorequicklythan does). Theaboveproportionalityconditionisalsosufficienttocompletely characterizetheexponentialdistribution. Proposition Theproportionality conditionis satisfiedonlyif hasanexponentialdistribution. Proof Theconditionalprobability canbewritten asDenote by thedistributionfunction of , that is,and by itssurvival function:Then,Dividing bothsidesby , we obtainwhere isaquantitythattendsto when tendsto . Takinglimitsonbothsides,we obtainor, bythedefinitionof derivative:This differentialequationiseasilysolvedbyusingthechain rule:Taking theintegralfrom to ofbothsides,we getandorBut (because cannottakenegativevalues) impliesExponentiating bothsides,we obtainTherefore,orBut thedensityfunctionisthefirstderivativeofthedistribution function:and therightmosttermisthedensityofanexponentialrandomvariable. Therefore,theproportionalityconditionissatisfiedonlyif isanexponentialrandomvariable Expectedvalue Theexpectedvalueofanexponentialrandom variable is Proof It canbederivedas follows: Variance Thevarianceofanexponentialrandomvariable is Proof It canbederivedthankstotheusual varianceformula (): Momentgeneratingfunction Themomentgeneratingfunctionofan exponentialrandomvariable isdefinedforany : Proof The definitionofmomentgeneratingfunction givesOf course,theaboveintegralsconvergeonlyif , i.e.onlyif . Therefore,themomentgeneratingfunctionofanexponentialrandomvariable existsforall . Characteristicfunction Thecharacteristicfunctionofanexponential randomvariable is Proof By usingthedefinitionofcharacteristicfunctionandthefactthat we can writeWe nowcomputeseparatelythetwointegrals.Thefirstintegral isTherefore,which canberearrangedto yieldorThe secondintegral isTherefore,which canberearrangedto yieldorBy puttingpiecestogether,we get Distributionfunction Thedistributionfunctionofanexponentialrandomvariable is Proof If , thenbecause cannottakeonnegativevalues.If , then Moredetails Inthefollowingsubsectionsyoucanfindmoredetailsabouttheexponential distribution. Memorylessproperty Oneofthemostimportantpropertiesoftheexponentialdistributionisthe memorylessproperty: for any . Proof Thisisprovedas follows: isthetimeweneedtowaitbeforeacertaineventoccurs.Theaboveproperty saysthattheprobabilitythattheeventhappensduringatimeintervalof length isindependentofhowmuchtimehasalreadyelapsed () withouttheeventhappening. Thesumofexponentialrandom variablesisaGammarandomvariable Supposethat , , ..., are mutuallyindependentrandomvariableshaving exponentialdistributionwithparameter . Define Then,thesum isaGammarandomvariablewithparameters and . Proof Thisisprovedusingmomentgenerating functions(rememberthatthemomentgeneratingfunctionofasumofmutually independentrandomvariablesisjusttheproductoftheirmomentgenerating functions):The latteristhemomentgeneratingfunctionofaGammadistributionwith parameters and . So hasaGammadistribution,becausetworandomvariableshavethesame distributionwhentheyhavethesamemomentgeneratingfunction. Therandomvariable isalsosometimessaidtohaveanErlangdistribution. TheErlangdistributionisjustaspecialcaseoftheGammadistribution:a GammarandomvariableisanErlangrandomvariableonlywhenitcanbewritten asasumofexponentialrandomvariables. RelationtothePoissondistribution TheexponentialdistributionisstrictlyrelatedtothePoissondistribution. Supposethat aneventcanoccurmorethanonce; thetimeelapsedbetweentwosuccessiveoccurrencesisexponentially distributedandindependentofpreviousoccurrences. Then,thenumberofoccurrencesoftheeventwithinagivenunitoftimehasa Poissondistribution. WeinvitethereadertoseethelectureonthePoisson distributionforamoredetailedexplanationandanintuitivegraphical representationofthisfact. Discretecounterpart Theexponentialdistributionisthecontinuouscounterpartofthe geometric distribution,whichisinsteaddiscrete. Densityplot Thenextplotshowshowthedensityoftheexponentialdistributionchangesby changingtherateparameter: thefirstgraph(redline)istheprobabilitydensityfunctionofan exponentialrandomvariablewithrateparameter ; thesecondgraph(blueline)istheprobabilitydensityfunctionofan exponentialrandomvariablewithrateparameter . Thethinverticallinesindicatethemeansofthetwodistributions.Note that,byincreasingtherateparameter,wedecreasethemeanofthe distributionfrom to . Solvedexercises Belowyoucanfindsomeexerciseswithexplainedsolutions. Exercise1 Let beanexponentialrandomvariablewithparameter . Computethefollowing probability: Solution Firstofallwecanwritetheprobability asusing thefactthattheprobabilitythatacontinuousrandomvariabletakesonany specificvalueisequaltozero(seeContinuous randomvariablesandzero-probabilityevents).Now,theprobabilitycanbe writtenintermsofthedistributionfunctionof as Exercise2 Supposetherandomvariable hasanexponentialdistributionwithparameter . Computethefollowing probability: Solution Thisprobabilitycanbeeasilycomputed byusingthedistributionfunctionof : Exercise3 Whatistheprobabilitythatarandomvariable islessthanitsexpectedvalue,if hasanexponentialdistributionwithparameter ? Solution Theexpectedvalueofanexponential randomvariablewithparameter isThe probabilityabovecanbecomputedbyusingthedistributionfunctionof : Howtocite Pleaseciteas: Taboga,Marco(2021)."Exponentialdistribution",Lecturesonprobabilitytheoryandmathematicalstatistics.KindleDirectPublishing.Onlineappendix.https://www.statlect.com/probability-distributions/exponential-distribution. Thebooks Mostofthelearningmaterialsfoundonthiswebsitearenowavailableinatraditionaltextbookformat. 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