Exponential Distribution (Definition, Formula ... - BYJU'S

In Probability theory and statistics, the exponential distribution is a continuous probability distribution that often concerns the amount of time until some ... MathsMathArticleExponentialDistribution Previous Next ExponentialDistribution Inprobabilitytheory,theexponentialdistributionisdefinedastheprobabilitydistributionoftimebetweeneventsinthePoissonpointprocess.Theexponentialdistributionisconsideredasaspecialcaseofthegammadistribution.Also,theexponentialdistributionisthecontinuousanalogueofthegeometricdistribution.Inthisarticle,wewilldiscusswhatisexponentialdistribution,itsformula,mean,variance,memorylesspropertyofexponentialdistribution,andsolvedexamples. TableofContents: WhatisExponentialDistribution? Formula MeanandVariance MemorylessProperty SumofTwoIndependentExponentialRandomVariables ExponentialDistributionGraph Applications Example FAQs WhatisExponentialDistribution? InProbabilitytheoryandstatistics,theexponentialdistributionisacontinuousprobabilitydistributionthatoftenconcernstheamountoftimeuntilsomespecificeventhappens.Itisaprocessinwhicheventshappencontinuouslyandindependentlyataconstantaveragerate.Theexponentialdistributionhasthekeypropertyofbeingmemoryless.Theexponentialrandomvariablecanbeeithermoresmallvaluesorfewerlargervariables.Forexample,theamountofmoneyspentbythecustomerononetriptothesupermarketfollowsanexponentialdistribution. ExponentialDistributionFormula Thecontinuousrandomvariable,sayXissaidtohaveanexponentialdistribution,ifithasthefollowingprobabilitydensityfunction: \(\begin{array}{l}f_{X}(x|\lambda)=\left\{\begin{matrix}\lambdae^{-\lambdax}&for\x>0\\0&for\x\leq0\end{matrix}\right.\end{array}\) Where λiscalledthedistributionrate. MeanandVarianceofExponentialDistribution Mean: Themeanoftheexponentialdistributioniscalculatedusingtheintegrationbyparts. Mean=E[X]=\(\begin{array}{l}\int_{0}^{\infty}x\lambdae^{-\lambdax}dx\end{array}\) \(\begin{array}{l}=\lambda\left[\left|\frac{-xe^{-\lambdax}}{\lambda}\right|^{\infty}_{0}+\frac{1}{\lambda}\int_{0}^{\infty}e^{-\lambdax}dx\right]\end{array}\) \(\begin{array}{l}=\lambda\left[0+\frac{1}{\lambda}\frac{-e^{-\lambdax}}{\lambda}\right]^{\infty}_{0}\end{array}\) \(\begin{array}{l}=\lambda\frac{1}{\lambda^{2}}\end{array}\) \(\begin{array}{l}=\frac{1}{\lambda}\end{array}\) Hence,themeanoftheexponentialdistributionis1/λ. Variance: Tofindthevarianceoftheexponentialdistribution,weneedtofindthesecondmomentoftheexponentialdistribution,anditisgivenby: \(\begin{array}{l}E[X^{2}]=\int_{0}^{\infty}x^{2}\lambdae^{-\lambdax}=\frac{2}{\lambda^{2}}\end{array}\) Hence,thevarianceofthecontinuousrandomvariable,Xiscalculatedas: Var(X)=E(X2)-E(X)2 Now,substitutingthevalueofmeanandthesecondmomentoftheexponentialdistribution,weget, Var(X)\(\begin{array}{l}=\frac{2}{\lambda^{2}}-\frac{1}{\lambda^{2}}=\frac{1}{\lambda^{2}}\end{array}\) Thus,thevarianceoftheexponentialdistributionis1/λ2. MemorylessPropertyofExponentialDistribution Themostimportantpropertyoftheexponentialdistributionisthememorylessproperty.Thispropertyisalsoapplicabletothegeometricdistribution. Anexponentiallydistributedrandomvariable“X”obeystherelation:  Pr(X>s+t|X>s)=Pr(X>t),foralls,t≥0 Now,letusconsiderthethecomplementarycumulativedistributionfunction: Pr(X>s+t|X>s) \(\begin{array}{l}=\frac{P_{r}(X>s+t\capX>s)}{P_{r}(X>s)}\end{array}\) \(\begin{array}{l}=\frac{P_{r}(X>s+t)}{P_{r}(X>s)}\end{array}\) \(\begin{array}{l}=\frac{e^{-\lambda(s+t)}}{e^{-\lambdas}}\end{array}\) =e-λt =Pr(X>t) Hence,Pr(X>s+t|X>s)=Pr(X>t) Thispropertyiscalledthememorylesspropertyoftheexponentialdistribution,aswedon’tneedtorememberwhentheprocesshasstarted. SumofTwoIndependentExponentialRandomVariables Theprobabilitydistributionfunctionofthetwoindependentrandomvariablesisthesumoftheindividualprobabilitydistributionfunctions.  IfX1andX2arethetwoindependentexponentialrandomvariableswithrespecttotherateparametersλ1andλ2respectively,thenthesumoftwoindependentexponentialrandomvariablesisgivenbyZ=X1+X2. \(\begin{array}{l}f_{Z}z=\int_{-\infty}^{\infty}f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})dx_{1}\end{array}\) \(\begin{array}{l}=\int_{0}^{z}\lambda_{1}e^{-\lambda_{1}x_{1}}\lambda_{2}e^{-\lambda_{2}(z-x_{1})}dx_{1}\end{array}\) \(\begin{array}{l}=\lambda_{1}\lambda_{2}e^{-\lambda_{2}z}\int_{0}^{z}e^{(\lambda_{2}-\lambda_{1})x_{1}}dx_{1}\end{array}\) \(\begin{array}{l}=\left\{\begin{matrix}\frac{\lambda_{1}\lambda_{2}}{\lambda_{2}-\lambda_{1}}(e^{-\lambda_{1}z}-e^{-\lambda_{2}z})&if\\lambda_{1}\neq\lambda_{2}\\\lambda^{2}ze^{-\lambdaz}&if\\lambda_{1}=\lambda_{2}=\lambda\end{matrix}\right.\end{array}\) ExponentialDistributionFormula ExponentialDistributionCalculator PoissonDistributionFormula ExponentialDistributionGraph Theexponentialdistributiongraphisagraphoftheprobabilitydensityfunctionwhichshowsthedistributionofdistanceortimetakenbetweenevents.Thetwotermsusedintheexponentialdistributiongraphislambda(λ)andx.Here,lambdarepresentstheeventsperunittimeandxrepresentsthetime.Thefollowinggraphshowsthevaluesfor λ=1and λ=2. ExponentialDistributionApplications Oneofthewidelyusedcontinuousdistributionistheexponentialdistribution.Ithelpstodeterminethetimeelapsedbetweentheevents.Itisusedinarangeofapplicationssuchasreliabilitytheory,queuingtheory,physicsandsoon.Someofthefieldsthataremodelledbytheexponentialdistributionareasfollows: ExponentialdistributionhelpstofindthedistancebetweenmutationsonaDNAstrand Calculatingthetimeuntiltheradioactiveparticledecays. Helpsonfindingtheheightofdifferentmoleculesinagasatthestabletemperatureandpressureinauniformgravitationalfield Helpstocomputethemonthlyandannualhighestvaluesofregularrainfallandriveroutflowvolumes ExponentialDistributionProblem Example: Assumethat,youusuallyget2phonecallsperhour.calculatetheprobability,thataphonecallwillcomewithinthenexthour. Solution: Itisgiventhat,2phonecallsperhour.So,itwouldexpectthatonephonecallateveryhalf-an-hour.So,wecantake λ=0.5 So,thecomputationisasfollows: \(\begin{array}{l}p(0\leqX\leq1)=\sum_{x=0}^{1}0.5e^{-0.5x}\end{array}\) =0.393469 Therefore,theprobabilityofarrivingthephonecallswithinthenexthouris 0.393469 StaytunedwithBYJU’S–TheLearningAppanddownloadtheapptolearnwitheasebyexploringmoreMaths-relatedvideos. FrequentlyAskedQuestionsonExponentialDistributionWhatismeantbyexponentialdistribution? Theexponentialdistributionisaprobabilitydistributionfunctionthatiscommonlyusedtomeasuretheexpectedtimeforaneventtohappen. WhatisthedifferencebetweenthePoissondistributionandexponentialdistribution? Poissondistributiondealswiththenumberofoccurrencesofeventsinafixedperiodoftime,whereastheexponentialdistributionisacontinuousprobabilitydistributionthatoftenconcernstheamountoftimeuntilsomespecificeventhappens. Whatisthemeanandthevarianceoftheexponentialdistribution? Themeanoftheexponentialdistributionis1/λandthevarianceoftheexponentialdistributionis1/λ2. Whyistheexponentialdistributionmemoryless? Thekeypropertyoftheexponentialdistributionismemorylessasthepasthasnoimpactonitsfuturebehaviour,andeachinstantislikethestartingofthenewrandomperiod. Whatdoeslambdameanintheexponentialdistribution? Thelambdainexponentialdistributionrepresentstherateparameter,anditdefinesthemeannumberofeventsinaninterval. 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