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| {{Distinguish2|the [[exponential family]] of probability distributions}}
| | My hobby is mainly Auto racing. Appears boring? Not!<br>I to learn Norwegian in my free time.<br><br>Here is my page; [http://Www.Sdruzeniantilopa.cz/groups/fifa-15-coin-hack/ FIFA coin generator] |
| <!-- EDITORS! Please see [[Wikipedia:WikiProject Probability#Standards]] for a discussion of standards used for probability distribution articles such as this one.-->
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| {{Probability distribution
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| | name = Exponential
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| | type = continuous
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| | pdf_image = [[File:exponential pdf.svg|325px|Probability density function]]
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| | cdf_image = [[File:exponential cdf.svg|325px|Cumulative distribution function]]
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| | parameters = {{nowrap|λ > 0}} rate, or inverse [[scale parameter|scale]]
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| | support = {{nowrap|''x'' ∈ [0, ∞)}}
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| | pdf = λ ''e''<sup>−λ''x''</sup>
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| | cdf = {{nowrap|1 − ''e''<sup>−λ''x''</sup>}}
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| | mean = λ<sup>−1</sup>
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| | median = {{nowrap|λ<sup>−1</sup> ln(2)}}
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| | mode = 0
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| | variance = λ<sup>−2</sup>
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| | skewness = 2
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| | kurtosis = 6
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| | entropy = {{nowrap|1 − ln(λ)}}
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| | mgf = <math>\left(1 - \frac{t}{\lambda}\right)^{-1}\, \text{ for } t < \lambda</math>
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| | char = <math>\left(1 - \frac{it}{\lambda}\right)^{-1}</math>
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| | rate = <math>\lambda z - \log(\lambda x) - 1,</math>
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| }}
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| In [[probability theory]] and [[statistics]], the '''exponential distribution''' (a.k.a. '''negative exponential distribution''') is the [[probability distribution]] that describes the time between events in a [[Poisson process]], i.e. a process in which events occur continuously and independently at a constant average rate. It is the continuous analogue of the [[geometric distribution]], and it has the key property of being [[memoryless]]. In addition to being used for the analysis of Poisson processes, it is found in various other contexts.
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| Note that the exponential distribution is not the same as the class of [[exponential family|exponential families]] of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the [[normal distribution]], [[binomial distribution]], [[gamma distribution]], [[Poisson distribution|Poisson]], and many others.
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| ==Characterization==
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| ===Probability density function===
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| The [[probability density function]] (pdf) of an exponential distribution is
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| :<math> f(x;\lambda) = \begin{cases}
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| \lambda e^{-\lambda x} & x \ge 0, \\
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| 0 & x < 0.
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| \end{cases}</math>
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| Alternatively, this can be defined using the [[Heaviside step function]], ''H''(''x'').
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| :<math>f(x;\lambda) = \mathrm \lambda e^{-\lambda x} H(x)</math>
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| Here λ > 0 is the parameter of the distribution, often called the ''rate parameter''. The distribution is supported on the interval [0, ∞). If a [[random variable]] ''X'' has this distribution, we write ''X'' ~ Exp(λ).
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| The exponential distribution exhibits [[infinite divisibility (probability)|infinite divisibility]].
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| ===Cumulative distribution function===
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| The [[cumulative distribution function]] is given by
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| :<math>F(x;\lambda) = \begin{cases}
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| 1-e^{-\lambda x} & x \ge 0, \\
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| 0 & x < 0.
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| \end{cases}</math>
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| Alternatively, this can be defined using the [[Heaviside step function]], ''H''(''x'').
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| :<math>F(x;\lambda) = \mathrm (1-e^{-\lambda x}) H(x)</math>
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| ===Alternative parameterization===
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| A commonly used alternative parameterization is to define the [[probability density function]] (pdf) of an exponential distribution as
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| :<math>f(x;\beta) = \begin{cases}
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| \frac{1}{\beta} e^{-\frac{x}{\beta}} & x \ge 0, \\
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| 0 & x < 0.
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| \end{cases}</math>
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| where β > 0 is a [[scale parameter]] of the distribution and is the [[multiplicative inverse|reciprocal]] of the ''rate parameter'', λ, defined above. In this specification, β is a ''survival parameter'' in the sense that if a [[random variable]] ''X'' is the duration of time that a given biological or mechanical system manages to survive and ''X'' ~ Exp(β) then E[''X''] = β. That is to say, the expected duration of survival of the system is β units of time. The parameterisation involving the "rate" parameter arises in the context of events arriving at a rate λ, when the time between events (which might be modelled using an exponential distribution) has a mean of β = λ<sup>−1</sup>.
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| The alternative specification is sometimes more convenient than the one given above, and some authors will use it as a standard definition. This alternative specification is not used here. Unfortunately this gives rise to a [[Mathematical notation|notational]] ambiguity. In general, the reader must check which of these two specifications is being used if an author writes "''X'' ~ Exp(λ)", since either the notation in the previous (using λ) or the notation in this section (here, using ''β'' to avoid confusion) could be intended.
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| ==Properties==
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| ===Mean, variance, moments and median===
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| [[File:Mean exp.svg|thumb|The mean is the probability mass centre, that is the [[first moment]].]]
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| [[File:Median exp.svg|thumb|The median is the [[preimage]] ''F''<sup>−1</sup>(1/2).]]
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| The mean or [[expected value]] of an exponentially distributed random variable ''X'' with rate parameter λ is given by
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| :<math>\mathrm{E}[X] = \frac{1}{\lambda}.</math>
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| In light of the examples given above, this makes sense: if you receive phone calls at an average rate of 2 per hour, then you can expect to wait half an hour for every call.
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| The [[variance]] of ''X'' is given by
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| :<math>\mathrm{Var}[X] = \frac{1}{\lambda^2},</math>
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| so the [[standard deviation]] is equal to the mean.
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| The [[Moment (mathematics)|moments]] of ''X'', for ''n'' = 1, 2, ..., are given by
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| :<math>\mathrm{E}\left [X^n \right ] = \frac{n!}{\lambda^n}.</math>
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| The [[median]] of ''X'' is given by
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| :<math>\text{m}[X] = \frac{\ln(2)}{\lambda} < \mathrm{E}[X],</math>
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| where ln refers to the [[natural logarithm]]. Thus the [[absolute difference]] between the mean and median is
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| :<math>|\text{E}[X]- \text{m}[X]| = \frac{1- \ln(2)}{\lambda}< \frac{1}{\lambda} = \text{standard deviation},</math>
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| in accordance with the [[Chebyshev's inequality#An application: distance between the mean and the median|median-mean inequality]].
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| ===Memorylessness===
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| An exponentially distributed random variable ''T'' obeys the relation
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| :<math>\Pr \left (T > s + t | T > s \right ) = \Pr(T > t), \qquad \forall s, t \ge 0. </math>
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| When ''T'' is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if ''T'' is conditioned on a failure to observe the event over some initial period of time ''s'', the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the [[conditional probability]] that occurrence will take at least 10 more seconds is equal to the unconditioned probability of observing the event more than 10 seconds relative to the initial time.
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| The exponential distributions and the [[geometric distribution]]s are the only memoryless probability distributions.
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| The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant [[Failure rate]].
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| ===Quantiles===
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| [[File:Tukey anomaly criteria for Exponential PDF.png|thumb|alt=Tukey anomaly criteria for exponential probability distribution function.| Tukey criteria for anomalies.<ref>{{cite journal |last1=Brillinger |first1= David R. |year=2011 |title=Data analysis, exploratory |journal= |volume= |issue= |pages=530–537 |publisher=SAGE Publications |doi= |url=http://www.stat.berkeley.edu/~brill/Papers/EDASage.pdf |accessdate=2013-11-21 }}</ref>]]
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| The [[quantile function]] (inverse cumulative distribution function) for Exp(λ) is
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| :<math>F^{-1}(p;\lambda) = \frac{-\ln(1-p)}{\lambda},\qquad 0 \le p < 1</math>
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| The [[quartile]]s are therefore:
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| *first quartile: ln(4/3)/λ
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| *[[median]]: ln(2)/λ
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| *third quartile: ln(4)/λ
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| And as a consequence the [[interquartile range]] is ln(3)/λ.
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| ===Kullback–Leibler divergence===
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| The directed [[Kullback–Leibler divergence]] of Exp(λ) ('approximating' distribution) from Exp(λ<sub>0</sub>) ('true' distribution) is given by
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| :<math>\Delta(\lambda_0 || \lambda) = \log(\lambda_0) - \log(\lambda) + \frac{\lambda}{\lambda_0} - 1.</math>
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| ===Maximum entropy distribution===
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| Among all continuous probability distributions with [[Support_%28mathematics%29#In_probability_and_measure_theory|support]] [0, ∞) and mean μ, the exponential distribution with λ = 1/μ has the largest [[differential entropy]]. In other words, it is the [[maximum entropy probability distribution]] for a random variate ''X'' for which E[''X''] is fixed and greater than zero.<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=Journal of Econometrics |volume= |issue= |pages=219–230 |publisher=Elsevier |doi= |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |accessdate=2011-06-02 }}</ref>
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| ===Distribution of the minimum of exponential random variables===
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| Let ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> be [[Independent random variables|independent]] exponentially distributed random variables with rate parameters λ<sub>1</sub>, ..., λ<sub>''n''</sub>. Then
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| :<math> \min \left \{X_1,\dots,X_n \right \} </math>
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| is also exponentially distributed, with parameter
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| :<math> \lambda = \lambda_1+\cdots+\lambda_n.</math>
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| This can be seen by considering the [[Cumulative distribution function#Complementary cumulative distribution function|complementary cumulative distribution function]]:
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| :<math> \begin{align}
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| \Pr \left (\min\{X_1,\dots,X_n \} > x \right ) & = \Pr\left(X_1 > x \wedge \cdots \wedge X_n > x\right) \\
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| &= \prod_{i=1}^n \Pr(X_i > x) \\
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| &= \prod_{i=1}^n \exp(-x\lambda_i) = \exp\left(-x\sum_{i=1}^n \lambda_i\right).
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| \end{align} </math>
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| The index of the variable which achieves the minimum is distributed according to the law
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| :<math>\Pr \left (X_k=\min\{X_1,\dots,X_n\} \right )=\frac{\lambda_k}{\lambda_1+\cdots+\lambda_n}.</math>
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| Note that
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| :<math> \max\{X_1,\dots,X_n\} </math>
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| is not exponentially distributed.
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| ==Parameter estimation==
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| Suppose a given variable is exponentially distributed and the rate parameter λ is to be estimated.
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| ===Maximum likelihood===
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| The [[likelihood function]] for λ, given an [[independent identically-distributed random variables|independent and identically distributed]] sample ''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) drawn from the variable, is:
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| :<math> L(\lambda) = \prod_{i=1}^n \lambda \exp(-\lambda x_i) = \lambda^n \exp \left(-\lambda \sum_{i=1}^n x_i\right)=\lambda^n\exp\left(-\lambda n \overline{x}\right), </math>
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| where:
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| :<math>\overline{x}={1 \over n}\sum_{i=1}^n x_i</math>
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| is the sample mean.
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| The derivative of the likelihood function's logarithm is:
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| :<math>\frac{\mathrm{d}}{\mathrm{d}\lambda} \ln (L(\lambda)) = \frac{\mathrm{d}}{\mathrm{d}\lambda} \left( n \ln(\lambda) - \lambda n\overline{x} \right) = \frac{n}{\lambda}-n\overline{x}\ \begin{cases} > 0 & 0 < \lambda < \frac{1}{\overline{x}}, \\[8pt] = 0 & \lambda = \frac{1}{\overline{x}}, \\[8pt] < 0 & \lambda > \frac{1}{\overline{x}}. \end{cases} </math>
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| Consequently the [[maximum likelihood]] estimate for the rate parameter is:
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| :<math>\widehat{\lambda} = \frac{1}{\overline{x}}.</math>
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| ===Confidence intervals===
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| The 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by:<ref>{{cite book|title=Introduction to probability and statistics for engineers and scientists|first=Sheldon M.|last=Ross|page=267|url=http://books.google.com/books?id=mXP_UEiUo9wC&pg=PA267| edition=4th|year=2009| publisher=Associated Press|isbn=978-0-12-370483-2}}</ref>
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| :<math>\frac{2n}{\widehat{\lambda} \chi^2_{1-\frac{\alpha}{2},2n}} < \frac{1}{\lambda} < \frac{2n}{\widehat{\lambda} \chi^2_{\frac{\alpha}{2},2n}}</math>
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| which is also equal to:
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| : <math>\frac{2n\overline{x}}{\chi^2_{1-\frac{\alpha}{2},2n}} < \frac{1}{\lambda} < \frac{2n\overline{x}}{\chi^2_{\frac{\alpha}{2},2n}}</math>
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| where {{math|χ<sup>2</sup><sub>''p'',''v''</sub>}} is the {{math|100(1 – ''p'')}} [[percentile]] of the [[chi squared distribution]] with ''v'' [[degrees of freedom (statistics)|degrees of freedom]], n is the number of observations of inter-arrival times in the sample, and x-bar is the sample average. A simple approximation to the exact interval endpoints can be derived using a normal approximation to the {{math|''χ''<sup>2</sup><sub>''p'',''v''</sub>}} distribution. This approximation gives the following values for a 95% confidence interval:
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| :<math> \lambda_{low}=\widehat{\lambda} \left (1-\frac{1.96}{\sqrt{n}} \right ) </math>
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| :<math> \lambda_{upp}=\widehat{\lambda} \left (1+\frac{1.96}{\sqrt{n}} \right ) </math>
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| This approximation may be acceptable for samples containing at least 15 to 20 elements.<ref name="Guerriero">{{Cite journal
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| | first1 = V.|last1= Guerriero | year = 2012 | title = Power Law Distribution: Method of Multi-scale Inferential Statistics| journal = Journal of Modern Mathematics Frontier (JMMF)| url =http://www.sjmmf.org/paperInfo.aspx?ID=886 | volume = 1 | pages = 21–28}}</ref>
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| ===Bayesian inference===
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| The [[conjugate prior]] for the exponential distribution is the [[gamma distribution]] (of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful:
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| :<math> \mathrm{Gamma}(\lambda; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda\beta). </math>
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| The [[posterior distribution]] ''p'' can then be expressed in terms of the likelihood function defined above and a gamma prior:
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| :<math> \begin{align}
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| p(\lambda) &\propto L(\lambda) \times \mathrm{Gamma}(\lambda; \alpha, \beta) \\
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| &= \lambda^n \exp\left (-\lambda n\overline{x} \right) \times \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda \beta) \\
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| &\propto \lambda^{(\alpha+n)-1} \exp(-\lambda \left (\beta + n\overline{x} \right)).
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| \end{align}</math>
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| Now the posterior density ''p'' has been specified up to a missing normalizing constant. Since it has the form of a gamma pdf, this can easily be filled in, and one obtains:
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| :<math> p(\lambda) = \mathrm{Gamma}(\lambda; \alpha + n, \beta + n \overline{x}). </math>
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| Here the parameter α can be interpreted as the number of prior observations, and β as the sum of the prior observations.
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| The posterior mean here is:
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| :<math> \frac{\alpha + n}{\beta + n \overline{x}}. </math>
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| ==Generating exponential variates==<!-- This section is linked from [[Gamma distribution]] -->
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| A conceptually very simple method for generating exponential [[random variate|variate]]s is based on [[inverse transform sampling method|inverse transform sampling]]: Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] on the unit interval (0, 1), the variate
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| :<math>T = F^{-1}(U)</math>
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| has an exponential distribution, where ''F''<sup> −1</sup> is the [[quantile function]], defined by
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| :<math>F^{-1}(p)=\frac{-\ln(1-p)}{\lambda}.</math>
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| Moreover, if ''U'' is uniform on (0, 1), then so is 1 − ''U''. This means one can generate exponential variates as follows:
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| :<math>T = \frac{-\ln(U)}{\lambda}.</math>
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| Other methods for generating exponential variates are discussed by Knuth<ref>[[Donald Knuth|Donald E. Knuth]] (1998). ''[[The Art of Computer Programming]]'', volume 2: ''Seminumerical Algorithms'', 3rd edn. Boston: Addison–Wesley. ISBN 0-201-89684-2. ''See section 3.4.1, p. 133.''</ref> and Devroye.<ref name="devroye">Luc Devroye (1986). ''[http://luc.devroye.org/rnbookindex.html Non-Uniform Random Variate Generation]''. New York: Springer-Verlag. ISBN 0-387-96305-7. ''See [http://luc.devroye.org/chapter_nine.pdf chapter IX], section 2, pp. 392–401.''</ref>
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| The [[ziggurat algorithm]] is a fast method for generating exponential variates.
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| A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.<ref name="devroye"/>
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| ==Related distributions==
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| {{More footnotes|section|date=March 2011}}
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| * Exponential distribution is closed under scaling by a positive factor. If ''X'' ~ Exp(λ) then ''kX'' ~ Exp(λ/''k'').
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| * If ''X'' ~ Exp(λ) and ''Y'' ~ Exp(ν) then min(''X'', ''Y'') ~ Exp(λ + ν).
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| * If ''X<sub>i</sub>'' ~ Exp(λ) then min{''X''<sub>1</sub>, ..., ''X<sub>n</sub>''} ~ Exp(''n''λ).
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| * The [[Benktander Weibull distribution]] reduces to a truncated exponential distribution. If ''X'' ~ Exp(λ) then 1+''X'' ~ [[Benktander Weibull distribution|BenktanderWeibull(λ, 1)]].
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| * The exponential distribution is a limit of a scaled [[beta distribution]]:
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| ::<math>\lim_{n \to \infty}n{\rm Beta}(1,n) = {\rm Exp}(1).</math>
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| * If ''X<sub>i</sub>'' ~ Exp(λ) then ''X''<sub>1</sub> + ... + ''X<sub>k</sub>'' ~ [[Erlang distribution|Erlang(''k'', λ)]]
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| * If ''X'' ~ Exp(1) then μ − σ log(''X'') ~ [[Generalized extreme value distribution|GEV(μ, σ, 0)]].
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| * If ''X'' ~ Exp(λ) then ''X'' ~ [[gamma distribution|Gamma(1, λ)]]
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| * If ''X'' ~ Exp(λ) and ''Y'' ~ Exp(ν) then λ''X'' − ν''Y'' ~ [[Laplace distribution|Laplace(0, 1)]].
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| * If ''X'', ''Y'' ~ Exp(λ) then ''X'' − ''Y'' ~ Laplace(0, λ<sup>−1</sup>).
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| * If ''X'' ~ Laplace(μ, β<sup>−1</sup>) then |''X'' − μ| ~ Exp(β).
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| * If ''X'' ~ Exp(1) then ([[logistic distribution]]):
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| ::<math>\mu-\beta\log \left(\tfrac{e^{-X}}{1-e^{-X}}\right) \sim \mathrm{Logistic}(\mu,\beta) </math>
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| * If ''X'', ''Y'' ~ Exp(1) then ([[logistic distribution]]):
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| ::<math>\mu-\beta\log\left(\tfrac{X}{Y}\right) \sim \mathrm{Logistic}(\mu,\beta) </math>
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| * If ''X'' ~ Exp(λ) then ''ke<sup>X</sup>'' ~ [[Pareto distribution|Pareto(''k'', λ)]].
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| * If ''X'' ~ Pareto(1, λ) then log(''X'') ~ Exp(λ).
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| * Exponential distribution is a special case of type 3 [[Pearson distribution]].
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| * If ''X'' ~ Exp(λ) then <math>\tfrac{e^{-X}}{k} \sim \mathrm{PowerLaw}(k, \lambda)</math> ([[power law]])
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| * If ''X'' ~ [[Rayleigh distribution|Rayleigh(λ<sup>−1/2</sup>)]] then ''X''<sup>2</sup>/2 ~ Exp(λ).
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| * If ''X'' ~ Exp(λ) then <math> X \sim \mathrm{Weibull}(\tfrac{1}{\lambda},1)</math> ([[Weibull distribution]])
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| * If ''X'' ~ Exp(1) then <math> \lambda X^{\tfrac{1}{k}} \sim \mathrm{Weibull}(\lambda,k)</math> ([[Weibull distribution]])
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| * If ''X<sub>i</sub>'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]] then
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| ::<math>\lim_{n \to \infty}n \min \left (X_1, \ldots, X_n \right ) \sim \textrm{Exp}(1)</math>
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| * If ''Y|X'' ~ [[Poisson distribution|Poisson(''X'')]] where ''X'' ~ Exp(λ<sup>−1</sup>) then <math>Y \sim \mathrm{Geometric}(\tfrac{1}{1+\lambda})</math> ([[geometric distribution]])
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| * If ''X'' ~ Exp(1) and <math>Y \sim \Gamma(\alpha,\tfrac{\beta}{\alpha})</math> then <math>\sqrt{XY} \sim \mathrm{K}(\alpha,\beta)</math> ([[K-distribution]])
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| * The [[Hoyt distribution]] can be obtained from Exponential distribution and [[Arcsine distribution]]
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| * If ''X'' ~ Exp(λ) and ''Y'' ~ Erlang(''n'', λ) then:
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| ::<math>\frac{X}{Y} \sim \mathrm{Pareto}(1,n)</math>
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| * If ''X'' ~ Exp(λ) and <math>Y \sim \Gamma(n,\tfrac{1}{\lambda})</math> then <math>\tfrac{X}{Y} \sim \mathrm{Pareto}(1,n)</math>
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| * If ''X'' ~ [[skew-logistic distribution|SkewLogistic(θ)]], then log(1 + ''e<sup>-X</sup>'') ~ Exp(θ).
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| * If ''X'' ~ Exp(λ) and {{nowrap|''Y'' {{=}} μ − β log(''X''λ)}} then ''Y'' ∼ [[Gumbel distribution|Gumbel(μ, β)]].
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| * If ''X'' ~ Exp(1/2) then {{nowrap|''X'' ∼ χ<sub>2</sub><sup>2</sup>}}, i.e. ''X'' has a [[chi-squared distribution]] with 2 [[degrees of freedom (statistics)|degrees of freedom]].
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| *Let {{nowrap|''X'' ∼ Exp(λ<sub>''X''</sub>)}} and {{nowrap|''Y'' ∼ Exp(λ<sub>''Y''</sub>)}} be independent. Then <math>Z = \frac{\lambda_X X}{\lambda_Y Y}</math> has probability density function <math>f_Z(z) = \frac{1}{(z + 1)^2}</math>. This can be used to obtain a [[confidence interval]] for <math>\frac{\lambda_X}{\lambda_Y}</math>.
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| Other related distributions:
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| *[[Hyper-exponential distribution]] – the distribution whose density is a weighted sum of exponential densities.
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| *[[Hypoexponential distribution]] – the distribution of a general sum of exponential random variables.
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| *[[exGaussian distribution]] – the sum of an exponential distribution and a [[normal distribution]].
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| ==Applications==
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| ===Occurrence of events===
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| The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous [[Poisson process]].
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| The exponential distribution may be viewed as a continuous counterpart of the [[geometric distribution]], which describes the number of [[Bernoulli trial]]s necessary for a ''discrete'' process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state.
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| In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives. Similar caveats apply to the following examples which yield approximately exponentially distributed variables:
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| * The time until a radioactive [[particle decay]]s, or the time between clicks of a [[geiger counter]]
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| * The time it takes before your next telephone call
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| * The time until default (on payment to company debt holders) in reduced form credit risk modeling
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| Exponential variables can also be used to model situations where certain events occur with a constant probability per unit length, such as the distance between [[mutation]]s on a [[DNA]] strand, or between [[roadkill]]s on a given road.{{Citation needed|date=March 2009}}
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| In [[queuing theory]], the service times of agents in a system (e.g. how long it takes for a bank teller etc. to serve a customer) are often modeled as exponentially distributed variables. (The inter-arrival of customers for instance in a system is typically modeled by the [[Poisson distribution]] in most management science textbooks.) The length of a process that can be thought of as a sequence of several independent tasks is better modeled by a variable following the [[Erlang distribution]] (which is the distribution of the sum of several independent exponentially distributed variables).
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| [[File:FitExponDistr.tif|thumb|260px|Fitted cumulative exponential distribution to annually maximum 1-day rainfalls using CumFreq<ref>{{cite web |url=http://www.waterlog.info/cumfreq.htm| title=Cumfreq, a free computer program for cumulative frequency analysis}}</ref>]]
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| [[Reliability theory]] and [[reliability engineering]] also make extensive use of the exponential distribution. Because of the ''[[#Memorylessness|memoryless]]'' property of this distribution, it is well-suited to model the constant [[hazard rate]] portion of the [[bathtub curve]] used in reliability theory. It is also very convenient because it is so easy to add [[failure rate]]s in a reliability model. The exponential distribution is however not appropriate to model the overall lifetime of organisms or technical devices, because the "failure rates" here are not constant: more failures occur for very young and for very old systems.
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| In [[physics]], if you observe a [[gas]] at a fixed [[temperature]] and [[pressure]] in a uniform [[gravitational field]], the heights of the various molecules also follow an approximate exponential distribution, known as the [[Barometric formula]]. This is a consequence of the entropy property mentioned below.
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| In [[hydrology]], the exponential distribution is used to analyze extreme values of such variables as monthly and annual maximum values of daily rainfall and river discharge volumes.<ref>{{cite book|last=Ritzema (ed.)|first=H.P.|title=Frequency and Regression Analysis|year=1994|publisher=Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands|pages=175–224|url=http://www.waterlog.info/pdf/freqtxt.pdf|isbn=90-70754-33-9}}</ref>
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| :The blue picture illustrates an example of fitting the exponential distribution to ranked annually maximum one-day rainfalls showing also the 90% [[confidence belt]] based on the [[binomial distribution]]. The rainfall data are represented by [[plotting position]]s as part of the [[cumulative frequency analysis]].
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| ===Prediction===
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| Having observed a sample of ''n'' data points from an unknown exponential distribution a common task is to use these samples to make predictions about future data from the same source. A common predictive distribution over future samples is the so-called plug-in distribution, formed by plugging a suitable estimate for the rate parameter λ into the exponential density function. A common choice of estimate is the one provided by the principle of maximum likelihood, and using this yields the predictive density over a future sample ''x''<sub>''n''+1</sub>, conditioned on the observed samples ''x'' = (''x''<sub>1</sub>, ..., ''x<sub>n</sub>'') given by
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| :<math>p_{\rm ML}(x_{n+1} \mid x_1, \ldots, x_n) = \left( \frac1{\overline{x}} \right) \exp \left( - \frac{x_{n+1}}{\overline{x}} \right)</math>
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| The Bayesian approach provides a predictive distribution which takes into account the uncertainty of the estimated parameter, although this may depend crucially on the choice of prior.
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| A predictive distribution free of the issues of choosing priors that arise under the subjective Bayesian approach is
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| :<math>p_{\rm CNML}(x_{n+1} \mid x_1, \ldots, x_n) = \frac{ n^{n+1} \left( \overline{x} \right)^n }{ \left( n \overline{x} + x_{n+1} \right)^{n+1} },</math>
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| which can be considered as
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| *(1) a frequentist [[confidence distribution]], obtained from the distribution of the pivotal quantity <math>{x_{n+1}}/{\overline{x}}</math>;<ref>Lawless, J.F., Fredette, M.,"Frequentist predictions intervals and predictive distributions", Biometrika (2005), Vol 92, Issue 3, pp 529–542.</ref>
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| *(2) a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of ''x''<sub>''n''+1</sub> and λ by maximization;<ref>Bjornstad, J.F., "Predictive Likelihood: A Review", Statist. Sci. Volume 5, Number 2 (1990), 242–254.</ref>
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| *(3) an objective Bayesian predictive posterior distribution, obtained using the non-informative [[Jeffreys prior]] 1/λ;
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| *(4) the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.<ref>D. F. Schmidt and E. Makalic, "[http://www.emakalic.org/blog/wp-content/uploads/2010/04/SchmidtMakalic09b.pdf Universal Models for the Exponential Distribution]", ''[[IEEE Transactions on Information Theory]]'', Volume 55, Number 7, pp. 3087–3090, 2009 {{doi|10.1109/TIT.2009.2018331}}</ref>
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| The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter, λ<sub>0</sub>, and the predictive distribution based on the sample ''x''. The [[Kullback–Leibler divergence]] is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(λ<sub>0</sub>||''p'') denote the Kullback–Leibler divergence between an exponential with rate parameter λ<sub>0</sub> and a predictive distribution ''p'' it can be shown that
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| :<math>\begin{align}
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| {\rm E}_{\lambda_0} \left[ \Delta(\lambda_0\mid\mid p_{\rm ML}) \right] &= \psi(n) + \frac{1}{n-1} - \log(n) \\
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| {\rm E}_{\lambda_0} \left[ \Delta(\lambda_0\mid\mid p_{\rm CNML}) \right] &= \psi(n) + \frac{1}{n} - \log(n)
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| \end{align}</math>
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| where the expectation is taken with respect to the exponential distribution with rate parameter {{nowrap|λ<sub>0</sub> ∈ (0, ∞)}}, and {{nowrap|ψ( · )}} is the digamma function. It is clear that the CNML predictive distribution is strictly superior to the maximum likelihood plug-in distribution in terms of average Kullback–Leibler divergence for all sample sizes {{nowrap|''n'' > 0}}.
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| ==See also==
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| * [[Dead time]] – an application of exponential distribution to particle detector analysis.
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| * [[Laplace distribution]], or the "double exponential distribution".
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| ==References==
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| {{Reflist}}
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| ==External links==
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| * {{springer|title=Exponential distribution|id=p/e036900}}
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| *[http://www.elektro-energetika.cz/calculations/ex.php?language=english Online calculator of Exponential Distribution]
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| {{ProbDistributions|continuous-semi-infinite}}
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| {{Common univariate probability distributions}}
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| {{DEFAULTSORT:Exponential Distribution}}
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| [[Category:Continuous distributions]]
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| [[Category:Exponentials]]
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| [[Category:Poisson processes]]
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| [[Category:Distributions with conjugate priors]]
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| [[Category:Exponential family distributions]]
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| [[Category:Infinitely divisible probability distributions]]
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| [[Category:Probability distributions]]
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