Editing Exponential distribution

{{pp|small=yes}}
{{Short description|Probability distribution}}
{{Distinguish|text=the [[exponential family]] of probability distributions}}
{{Probability distribution
  | name       = Exponential
  | type       = continuous
  | pdf_image  = [[File:Exponential distribution pdf - public domain.svg|325px|plot of the probability density function of the exponential distribution]]
  | cdf_image  = [[File:Exponential distribution cdf - public domain.svg|325px|Cumulative distribution function]]
  | parameters = <math>\lambda > 0,</math> rate, or inverse [[scale parameter|scale]]
  | support    = <math>x \in [0, \infty)</math>
  | pdf        = <math>\lambda e^{-\lambda x}</math>
  | cdf        = <math>1 - e^{-\lambda x}</math>
  | quantile   = <math>-\frac{\ln(1 - p)}{\lambda}</math>
  | mean       = <math>\frac{1}{\lambda}</math>
  | median     = <math>\frac{\ln 2}{\lambda}</math>
  | mode       = <math>0</math>
  | variance   = <math> \frac{1}{\lambda^2}</math>
  | skewness   = <math>2</math>
  | kurtosis   = <math>6</math>
  | entropy    = <math>1 - \ln\lambda</math>
  | mgf        = <math>\frac{\lambda}{\lambda-t}, \text{ for } t < \lambda</math>
  | char       = <math>\frac{\lambda}{\lambda-it}</math>
  | fisher     = <math>\frac{1}{\lambda^2}</math>
  | KLDiv      = <math>\ln\frac{\lambda_0}{\lambda} + \frac{\lambda}{\lambda_0} - 1</math>
  |ES=<math>\frac{-\ln(1 - p) + 1}{\lambda}</math>|bPOE=<math>e^{1-\lambda x}</math>}}

In [[probability theory]] and [[statistics]], the '''exponential distribution''' or '''negative exponential distribution''' is the [[probability distribution]] of the distance between events in a [[Poisson point process]], i.e., a process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process.<ref>{{Cite web |date=2021-07-15 |title=7.2: Exponential Distribution |url=https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Inferential_Statistics_and_Probability_-_A_Holistic_Approach_(Geraghty)/07:_Continuous_Random_Variables/7.02:_Exponential_Distribution |access-date=2024-10-11 |website=Statistics LibreTexts |language=en}}</ref> It is a particular case of the [[gamma distribution]]. It is the continuous analogue of the [[geometric distribution]], and it has the key property of being [[memoryless]].<ref>{{Cite web |title=Exponential distribution {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/exponential-distribution |access-date=2024-10-11 |website=www.britannica.com |language=en}}</ref> In addition to being used for the analysis of Poisson point processes it is found in various other contexts.<ref name="Weisstein">{{Cite web |last=Weisstein |first=Eric W. |title=Exponential Distribution |url=https://mathworld.wolfram.com/ExponentialDistribution.html |access-date=2024-10-11 |website=mathworld.wolfram.com |language=en}}</ref>

The exponential distribution is not the same as the class of [[exponential families]] of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like the [[normal distribution|normal]], [[binomial distribution|binomial]], [[gamma distribution|gamma]], and [[Poisson distribution|Poisson]] distributions.<ref name="Weisstein" />

==Definitions==

===Probability density function===
The [[probability density function]] (pdf) of an exponential distribution is

:<math> f(x;\lambda) = \begin{cases}
\lambda  e^{ - \lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>

Here ''λ'' > 0 is the parameter of the distribution, often called the ''rate parameter''. The distribution is supported on the interval&nbsp;{{closed-open|0, ∞}}. If a [[random variable]] ''X'' has this distribution, we write&nbsp;{{math|''X'' ~ Exp(''λ'')}}.

The exponential distribution exhibits [[infinite divisibility (probability)|infinite divisibility]].

===Cumulative distribution function===
The [[cumulative distribution function]] is given by

:<math>F(x;\lambda) = \begin{cases}
1-e^{-\lambda x} & x \ge 0, \\
0 & x < 0.
\end{cases}</math>

===Alternative parametrization===
The exponential distribution is sometimes parametrized in terms of the [[scale parameter]] {{math|1=''β'' = 1/''λ''}}, which is also the mean:
<math display="block">f(x;\beta) = \begin{cases}
   \frac{1}{\beta} e^{-x/\beta} & x \ge 0, \\
    0 & x < 0.
 \end{cases} \qquad\qquad
F(x;\beta) = \begin{cases}
   1- e^{-x/\beta} & x \ge 0, \\
    0 & x < 0.
 \end{cases} </math>

==Properties==

===Mean, variance, moments, and median===
[[File:Mean exp.svg|thumb|The mean is the probability mass centre, that is, the [[first moment]].]]
[[File:Median exp.svg|thumb|The median is the [[preimage]] ''F''<sup>−1</sup>(1/2).]]

The mean or [[expected value]] of an exponentially distributed random variable ''X'' with rate parameter ''λ'' is given by
<math display="block">\operatorname{E}[X] = \frac{1}{\lambda}.</math>

In light of the examples given [[#Occurrence and applications|below]], this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes.

The [[variance]] of ''X'' is given by
<math display="block">\operatorname{Var}[X] = \frac{1}{\lambda^2},</math>
so the [[standard deviation]] is equal to the mean.

The [[Moment (mathematics)|moments]] of ''X'', for <math>n\in\N</math> are given by
<math display="block">\operatorname{E}\left[X^n\right] = \frac{n!}{\lambda^n}.</math>

The [[central moment]]s of ''X'', for <math>n\in\N</math> are given by
<math display="block">\mu_n = \frac{!n}{\lambda^n} = \frac{n!}{\lambda^n}\sum^n_{k=0}\frac{(-1)^k}{k!}.</math>
where !''n'' is the [[subfactorial]] of ''n''

The [[median]] of ''X'' is given by
<math display="block">\operatorname{m}[X] = \frac{\ln(2)}{\lambda} < \operatorname{E}[X],</math>
where {{math|ln}} refers to the [[natural logarithm]].  Thus the [[absolute difference]] between the mean and median is
<math display="block">\left|\operatorname{E}\left[X\right] - \operatorname{m}\left[X\right]\right| = \frac{1 - \ln(2)}{\lambda} < \frac{1}{\lambda} = \operatorname{\sigma}[X],</math>

in accordance with the [[median-mean inequality]].

===Memorylessness property of exponential random variable===
An exponentially distributed random variable ''T'' obeys the relation
<math display="block">\Pr \left (T > s + t \mid T > s \right ) = \Pr(T > t), \qquad \forall s, t \ge 0.</math>

This can be seen by considering the [[complementary cumulative distribution function]]:
<math display="block"> \begin{align}
  \Pr\left(T > s + t \mid T > s\right)
    &= \frac{\Pr\left(T > s + t \cap T > s\right)}{\Pr\left(T > s\right)} \\[4pt]
    &= \frac{\Pr\left(T > s + t \right)}{\Pr\left(T > s\right)} \\[4pt]
    &= \frac{e^{-\lambda(s + t)}}{e^{-\lambda s}} \\[4pt]
    &= e^{-\lambda t} \\[4pt]
    &= \Pr(T > t).
\end{align} </math>

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 unconditional probability of observing the event more than 10 seconds after the initial time.

The exponential distribution and the [[geometric distribution]] are [[memorylessness|the only memoryless probability distributions]].

The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant [[failure rate]].

===Quantiles===
[[File:Tukey anomaly criteria for Exponential PDF.png|500px|thumb|alt=Tukey anomaly criteria for exponential probability distribution function.| Tukey criteria for anomalies.{{citation needed|date=September 2017}}]]

The [[quantile function]] (inverse cumulative distribution function) for Exp(''λ'') is
<math display="block">F^{-1}(p;\lambda) = \frac{-\ln(1-p)}{\lambda},\qquad 0 \le p < 1</math>

The [[quartile]]s are therefore:
* first quartile: ln(4/3)/''λ''
* [[median]]: ln(2)/''λ''
* third quartile: ln(4)/''λ''

And as a consequence the [[interquartile range]] is ln(3)/''λ''.

===Conditional Value at Risk (Expected Shortfall)===
The conditional value at risk (CVaR) also known as the [[expected shortfall]] or superquantile for Exp(''λ'') is derived as follows:<ref name="Norton-2019">{{cite journal |last1=Norton |first1=Matthew |last2=Khokhlov |first2=Valentyn |last3=Uryasev |first3=Stan |year=2019 |title=Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation |journal=Annals of Operations Research |volume=299 |issue=1–2 |pages=1281–1315 |publisher=Springer |doi=10.1007/s10479-019-03373-1 |arxiv=1811.11301 |url=http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |access-date=2023-02-27 |archive-date=2023-03-31 |archive-url=https://web.archive.org/web/20230331230821/http://uryasev.ams.stonybrook.edu/wp-content/uploads/2019/10/Norton2019_CVaR_bPOE.pdf |url-status=dead }}</ref>

<math display="block">\begin{align}
\bar{q}_\alpha (X) &=  \frac{1}{1-\alpha} \int_{\alpha}^{1} q_p (X) dp \\
&=  \frac{1}{(1-\alpha)} \int_{\alpha}^{1} \frac{-\ln (1 - p )}{\lambda}  dp \\
&=  \frac{-1}{\lambda(1-\alpha)} \int_{1-\alpha}^{0} -\ln (y )  dy \\
&=  \frac{-1}{\lambda(1-\alpha)} \int_{0}^{1 - \alpha} \ln (y )  dy \\
&=   \frac{-1}{\lambda(1-\alpha)}  [  ( 1-\alpha) \ln(1-\alpha) - (1-\alpha) ] \\
&=   \frac{ - \ln(1-\alpha) + 1 } { \lambda} \\
\end{align}
</math>

===Buffered Probability of Exceedance (bPOE)===
{{Main|Buffered probability of exceedance}}
The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold <math>x</math>.  It is derived as follows:<ref name="Norton-2019" />

<math display="block">\begin{align}
\bar{p}_x (X) &=  \{ 1 - \alpha | \bar{q}_\alpha (X) = x \} \\
&=  \{ 1 - \alpha |\frac{ - \ln(1-\alpha) + 1 } { \lambda} = x \} \\
&=  \{ 1 - \alpha |  \ln(1-\alpha)  =  1-\lambda x \} \\
&=  \{ 1 - \alpha |  e^{\ln(1-\alpha)}  =  e^{1-\lambda x} \} =  \{ 1 - \alpha |  1-\alpha =  e^{1-\lambda x} \} = e^{1-\lambda x}
\end{align}
</math>

===Kullback–Leibler divergence===
The directed [[Kullback–Leibler divergence]] in [[nat (unit)|nats]] of <math>e^\lambda</math> ("approximating" distribution) from <math>e^{\lambda_0}</math> ('true' distribution) is given by
<math display="block">\begin{align}
\Delta(\lambda_0 \parallel \lambda)
&= \mathbb{E}_{\lambda_0}\left( \log \frac{p_{\lambda_0}(x)}{p_\lambda(x)}\right)\\
&= \mathbb{E}_{\lambda_0}\left( \log \frac{\lambda_0 e^{\lambda_0 x}}{\lambda e^{\lambda x}}\right)\\
&= \log(\lambda_0) - \log(\lambda) - (\lambda_0 - \lambda)E_{\lambda_0}(x)\\
&= \log(\lambda_0) - \log(\lambda) + \frac{\lambda}{\lambda_0} - 1.
\end{align}
</math>

===Maximum entropy distribution===
Among all continuous probability distributions with [[Support (mathematics)#In probability and measure theory|support]] {{closed-open|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'' which is greater than or equal to zero and for which E[''X''] is fixed.<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=150 |issue=2 |pages=219–230 |publisher=Elsevier |doi=10.1016/j.jeconom.2008.12.014 |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |access-date=2011-06-02 |archive-url=https://web.archive.org/web/20160307144515/http://wise.xmu.edu.cn/uploadfiles/paper-masterdownload/2009519932327055475115776.pdf |archive-date=2016-03-07 |url-status=dead }}</ref>

===Distribution of the minimum of exponential random variables===
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
<math display="block">\min\left\{X_1, \dotsc, X_n \right\}</math>
is also exponentially distributed, with parameter
<math display="block">\lambda = \lambda_1 + \dotsb + \lambda_n.</math>

This can be seen by considering the [[complementary cumulative distribution function]]:
<math display="block">\begin{align}
      &\Pr\left(\min\{X_1, \dotsc, X_n\} > x\right) \\
  ={} &\Pr\left(X_1 > x, \dotsc, X_n > x\right) \\
  ={} &\prod_{i=1}^n \Pr\left(X_i > x\right) \\
  ={} &\prod_{i=1}^n \exp\left(-x\lambda_i\right) = \exp\left(-x\sum_{i=1}^n \lambda_i\right).
\end{align}</math>

The index of the variable which achieves the minimum is distributed according to the categorical distribution
<math display="block">\Pr\left(X_k = \min\{X_1, \dotsc, X_n\}\right) = \frac{\lambda_k}{\lambda_1 + \dotsb + \lambda_n}.</math>

A proof can be seen by letting <math>I = \operatorname{argmin}_{i \in \{1, \dotsb, n\}}\{X_1, \dotsc, X_n\}</math>. Then,
<math display="block">\begin{align}
\Pr (I = k) &= \int_{0}^{\infty} \Pr(X_k = x) \Pr(\forall_{i\neq k}X_{i} > x ) \,dx \\
      &= \int_{0}^{\infty} \lambda_k e^{- \lambda_k x} \left(\prod_{i=1, i\neq k}^{n} e^{- \lambda_i x}\right) dx \\
      &= \lambda_k \int_{0}^{\infty}  e^{- \left(\lambda_1 + \dotsb  +\lambda_n\right) x}  dx \\
      &= \frac{\lambda_k}{\lambda_1 + \dotsb + \lambda_n}.
\end{align}</math>

Note that
<math display="block">\max\{X_1, \dotsc, X_n\}</math>
is not exponentially distributed, if ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> do not all have parameter 0.<ref>{{cite web|last1=Michael|first1=Lugo|title=The expectation of the maximum of exponentials| url=http://www.stat.berkeley.edu/~mlugo/stat134-f11/exponential-maximum.pdf|access-date=13 December 2016|archive-url=https://web.archive.org/web/20161220132822/https://www.stat.berkeley.edu/~mlugo/stat134-f11/exponential-maximum.pdf |archive-date=20 December 2016|url-status=dead}}</ref>

===Joint moments of i.i.d. exponential order statistics===

Let <math> X_1, \dotsc, X_n </math> be <math> n </math> [[independent and identically distributed]] exponential random variables with rate parameter ''λ''.
Let <math> X_{(1)}, \dotsc, X_{(n)} </math> denote the corresponding [[order statistic]]s.
For <math> i < j </math> , the joint moment <math> \operatorname E\left[X_{(i)} X_{(j)}\right] </math> of the order statistics <math> X_{(i)} </math> and <math> X_{(j)} </math> is given by
<math display="block">\begin{align}
  \operatorname E\left[X_{(i)} X_{(j)}\right]
    &= \sum_{k=0}^{j-1}\frac{1}{(n - k)\lambda} \operatorname E\left[X_{(i)}\right] + \operatorname E\left[X_{(i)}^2\right] \\
    &= \sum_{k=0}^{j-1}\frac{1}{(n - k)\lambda}\sum_{k=0}^{i-1}\frac{1}{(n - k)\lambda} + \sum_{k=0}^{i-1}\frac{1}{((n - k)\lambda)^2} + \left(\sum_{k=0}^{i-1}\frac{1}{(n - k)\lambda}\right)^2.
\end{align}</math>

This can be seen by invoking the [[law of total expectation]] and the memoryless property:
<math display="block">\begin{align}
  \operatorname E\left[X_{(i)} X_{(j)}\right]
    &= \int_0^\infty \operatorname E\left[X_{(i)} X_{(j)} \mid X_{(i)}=x\right] f_{X_{(i)}}(x) \, dx \\
    &= \int_{x=0}^\infty x \operatorname E\left[X_{(j)} \mid X_{(j)} \geq x\right] f_{X_{(i)}}(x) \, dx
      &&\left(\textrm{since}~X_{(i)} = x \implies X_{(j)} \geq x\right) \\
    &= \int_{x=0}^\infty x \left[ \operatorname E\left[X_{(j)}\right] + x \right] f_{X_{(i)}}(x) \, dx
      &&\left(\text{by the memoryless property}\right) \\
    &= \sum_{k=0}^{j-1}\frac{1}{(n - k)\lambda} \operatorname E\left[X_{(i)}\right] + \operatorname E\left[X_{(i)}^2\right].
\end{align}</math>

The first equation follows from the [[law of total expectation]].
The second equation exploits the fact that once we condition on <math> X_{(i)} = x </math>, it must follow that <math> X_{(j)} \geq x </math>. The third equation relies on the memoryless property to replace <math>\operatorname E\left[ X_{(j)} \mid X_{(j)} \geq x\right]</math> with <math>\operatorname E\left[X_{(j)}\right] + x</math>.

===Sum of two independent exponential random variables===
The probability distribution function (PDF) of a sum of two independent random variables is the [[convolution of probability distributions|convolution of their individual PDFs]].  If <math>X_1</math> and <math>X_2</math> are independent exponential random variables with respective rate parameters <math>\lambda_1</math> and <math>\lambda_2,</math> then the probability density of <math>Z=X_1+X_2</math> is given by
<math display="block"> \begin{align}
 f_Z(z) &= \int_{-\infty}^\infty f_{X_1}(x_1) f_{X_2}(z - x_1)\,dx_1\\
   &= \int_0^z \lambda_1 e^{-\lambda_1 x_1} \lambda_2 e^{-\lambda_2(z - x_1)} \, dx_1 \\
   &= \lambda_1 \lambda_2 e^{-\lambda_2 z} \int_0^z e^{(\lambda_2 - \lambda_1)x_1}\,dx_1 \\
   &= \begin{cases}
        \dfrac{\lambda_1 \lambda_2}{\lambda_2-\lambda_1} \left(e^{-\lambda_1 z} - e^{-\lambda_2 z}\right) & \text{ if } \lambda_1 \neq \lambda_2 \\[4 pt]
        \lambda^2 z e^{-\lambda z} & \text{ if } \lambda_1 = \lambda_2 = \lambda.
      \end{cases}
 \end{align} </math>
The entropy of this distribution is available in closed form: assuming <math>\lambda_1 > \lambda_2</math> (without loss of generality), then
<math display="block">\begin{align}
 H(Z) &= 1 + \gamma + \ln \left( \frac{\lambda_1 - \lambda_2}{\lambda_1 \lambda_2} \right) + \psi \left( \frac{\lambda_1}{\lambda_1 - \lambda_2} \right) ,
\end{align}</math>
where <math>\gamma</math> is the [[Euler-Mascheroni constant]], and <math>\psi(\cdot)</math> is the [[digamma function]].<ref>{{cite arXiv|last1=Eckford |first1=Andrew W. |last2=Thomas |first2=Peter J. |date=2016 |title=Entropy of the sum of two independent, non-identically-distributed exponential random variables |class=cs.IT |eprint=1609.02911}}</ref>

In the case of equal rate parameters, the result is an [[Erlang distribution]] with shape 2 and parameter <math>\lambda,</math> which in turn is a special case of [[gamma distribution]].

The sum of n independent Exp(''λ)'' exponential random variables is Gamma(n, ''λ)'' distributed.

==Related distributions==
* If ''X'' ~ [[Laplace distribution|Laplace(μ, β<sup>−1</sup>)]], then |''X'' − μ| ~ Exp(β).<ref name="Leemis" />
* If ''X'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]] then −log(''X'') ~ Exp(1).
* If ''X'' ~ [[Pareto distribution|Pareto(1, λ)]], then log(''X'') ~ Exp(λ).<ref name="Leemis">{{cite journal|title=Univariate Distribution Relationships|first1=Lawrence M.|last1=Leemis|first2=Jacquelyn T.|last2=McQuestion|journal=The American Statistician|date=February 2008|volume=62|number=1|page=45-53|doi=10.1198/000313008X270448 |url=https://www.math.wm.edu/~leemis/2008amstat.pdf}}</ref>
* If ''X'' ~ [[Skew-logistic distribution|SkewLogistic(θ)]], then <math>\log\left(1 + e^{-X}\right) \sim \operatorname{Exp}(\theta)</math>.
* If ''X<sub>i</sub>'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]] then <math display="block">\lim_{n \to \infty}n \min \left(X_1, \ldots, X_n\right) \sim \operatorname{Exp}(1)</math>
* The exponential distribution is a limit of a scaled [[beta distribution]]: <math display="block">\lim_{n \to \infty} n \operatorname{Beta}(1, n) = \operatorname{Exp}(1).</math>
* The exponential distribution is a special case of type 3 [[Pearson distribution]].
* The exponential distribution is the special case of a [[Gamma distribution]] with shape parameter 1.<ref name="Leemis" />
* If ''X'' ~ Exp(λ) and ''X''{{sub|''i''}} ~ Exp(λ{{sub|''i''}}) then:
** <math>kX \sim \operatorname{Exp}\left(\frac{\lambda}{k}\right)</math>, closure under scaling by a positive factor.
** 1&nbsp;+&nbsp;''X'' ~ [[Benktander Weibull distribution|BenktanderWeibull]](λ, 1), which reduces to a truncated exponential distribution.
** ''ke<sup>X</sup>'' ~ [[Pareto distribution|Pareto]](''k'', λ).<ref name="Leemis" />
** ''e<sup>−λX</sup>'' ~ [[Uniform distribution (continuous)|''U''(0, 1)]].
** ''e<sup>−X</sup>'' ~ [[Beta distribution|Beta]](λ, 1).<ref name="Leemis" />
** {{sfrac|1|k}}''e''{{sup|''X''}} ~ [[power law|PowerLaw]](''k'', λ)
** <math>\sqrt{X} \sim \operatorname{Rayleigh} \left(\frac{1}{\sqrt{2\lambda}}\right)</math>, the [[Rayleigh distribution]]<ref name="Leemis" />
** <math>X \sim \operatorname{Weibull}\left(\frac{1}{\lambda}, 1\right)</math>, the [[Weibull distribution]]<ref name="Leemis" />
** <math>X^2 \sim \operatorname{Weibull}\left(\frac{1}{\lambda^2}, \frac{1}{2}\right)</math><ref name="Leemis" />
** {{nowrap|μ − β log(λ''X'') ∼ [[Gumbel distribution|Gumbel]](μ, β)}}.
** <math>\lfloor X\rfloor \sim \operatorname{Geometric}\left(1-e^{-\lambda}\right)</math>, a [[geometric distribution]] on 0,1,2,3,...
** <math>\lceil X\rceil \sim \operatorname{Geometric}\left(1-e^{-\lambda}\right)</math>, a [[geometric distribution]] on 1,2,3,4,...
** If also ''Y'' ~ Erlang(''n'', λ) or<math>Y \sim \Gamma\left(n, \frac{1}{\lambda}\right)</math> then <math>\frac{X}{Y} + 1 \sim \operatorname{Pareto}(1, n)</math>
** If also λ ~ [[gamma distribution|Gamma]](''k'', θ) (shape, scale parametrisation) then the marginal distribution of ''X'' is [[Lomax distribution|Lomax]](''k'', 1/θ), the gamma [[compound distribution|mixture]]
** λ{{sub|1}}''X''{{sub|1}} − λ{{sub|2}}''Y''{{sub|2}} ~ [[Laplace distribution|Laplace(0, 1)]].
** min{''X''<sub>1</sub>, ..., ''X<sub>n</sub>''} ~ Exp(λ<sub>1</sub> + ... + λ<sub>''n''</sub>).
** If also λ{{sub|''i''}} = λ then:
*** <math>X_1 + \cdots + X_k = \sum_i X_i \sim</math> [[Erlang distribution|Erlang]](''k'', λ) = [[gamma distribution|Gamma]](''k'', λ) with integer shape parameter ''k'' and rate parameter λ.<ref>{{cite book| title=Fundamentals of Applied Probability and Random Processes|first=Oliver C.|last=Ibe| page=128| url=https://books.google.com/books?id=K10XAwAAQBAJ| edition=2nd|year=2014| publisher=Academic Press| isbn=9780128010358}}</ref>
*** If <math>T = (X_1 + \cdots + X_n ) = \sum_{i=1}^n X_i</math>, then <math>2 \lambda T \sim \chi^2_{2n}</math>.
*** ''X''{{sub|''i''}} − ''X''{{sub|''j''}} ~ Laplace(0, λ<sup>−1</sup>).
** If also ''X''{{sub|''i''}} are independent, then:
*** <math>\frac{X_i}{X_i + X_j}</math> ~ [[uniform distribution (continuous)|U]](0, 1)
*** <math>Z = \frac{\lambda_i X_i}{\lambda_j X_j}</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_i}{\lambda_j}</math>.
** If also λ = 1:
*** <math>\mu - \beta\log\left(\frac{e^{-X}}{1 - e^{-X}}\right) \sim \operatorname{Logistic}(\mu, \beta)</math>, the [[logistic distribution]]
*** <math>\mu - \beta\log\left(\frac{X_i}{X_j}\right) \sim \operatorname{Logistic}(\mu, \beta)</math>
*** ''μ'' − σ log(''X'') ~ [[generalized extreme value distribution|GEV(μ, σ, 0)]].
*** Further if <math>Y \sim \Gamma\left(\alpha, \frac{\beta}{\alpha}\right)</math> then <math>\sqrt{XY} \sim \operatorname{K}(\alpha, \beta)</math> ([[K-distribution]])
** If also λ = 1/2 then {{nowrap|''X'' ∼ χ{{su|b=2|p=2}}}}; i.e., ''X'' has a [[chi-squared distribution]] with 2 [[degrees of freedom (statistics)|degrees of freedom]]. Hence: <math display="block">\operatorname{Exp}(\lambda) = \frac{1}{2\lambda} \operatorname{Exp}\left(\frac{1}{2} \right) \sim \frac{1}{2\lambda} \chi_2^2\Rightarrow \sum_{i=1}^n \operatorname{Exp}(\lambda) \sim \frac{1}{2\lambda }\chi_{2n}^2</math>
* If <math>X \sim \operatorname{Exp}\left(\frac{1}{\lambda}\right)</math> and <math>Y \mid X</math> ~ [[Poisson distribution|Poisson(''X'')]] then <math>Y \sim \operatorname{Geometric}\left(\frac{1}{1 + \lambda}\right)</math>     ([[geometric distribution]])
* The [[Hoyt distribution]] can be obtained from exponential distribution and [[arcsine distribution]]
* The exponential distribution is a limit of the [[Kaniadakis Exponential distribution|''κ''-exponential distribution]] in the <math>\kappa = 0</math> case.
* Exponential distribution is a limit of the [[κ-Generalized Gamma distribution]] in the <math>\alpha = 1</math> and <math>\nu = 1</math> cases:
*: <math>\lim_{(\alpha,\nu)\to(0,1)} p_\kappa(x) = (1+\kappa\nu)(2\kappa)^\nu \frac{\Gamma\Big(\frac{1}{2\kappa}+\frac{\nu}{2}\Big)}{\Gamma\Big(\frac{1}{2\kappa}-\frac{\nu}{2}\Big)} \frac{\alpha \lambda^\nu}{\Gamma(\nu)} x^{\alpha\nu-1}\exp_\kappa(-\lambda x^\alpha) = \lambda  e^{ - \lambda x} </math>

Other related distributions:
* [[Hyper-exponential distribution]] – the distribution whose density is a weighted sum of exponential densities.
* [[Hypoexponential distribution]] – the distribution of a general sum of exponential random variables.<ref name="Leemis" />
* [[exGaussian distribution]] – the sum of an exponential distribution and a [[normal distribution]].

==Statistical inference==
Below, suppose random variable ''X'' is exponentially distributed with rate parameter λ, and <math>x_1, \dotsc, x_n</math> are ''n'' independent samples from ''X'', with sample mean <math>\bar{x}</math>.

===Parameter estimation===

The [[maximum likelihood]] estimator for λ is constructed as follows.

The [[likelihood function]] for λ, given an [[independent and identically distributed]] sample ''x'' = (''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub>) drawn from the variable, is:
<math display="block"> 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>

where:
<math display="block">\overline{x} = \frac{1}{n}\sum_{i=1}^n x_i</math>
is the sample mean.

The derivative of the likelihood function's logarithm is:
<math display="block">
  \frac{d}{d\lambda} \ln L(\lambda) =
  \frac{d}{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>

Consequently, the [[maximum likelihood]] estimate for the rate parameter is:
<math display="block">\widehat{\lambda}_\text{mle} = \frac{1}{\overline{x}} = \frac{n}{\sum_i x_i}</math>

This is {{em|not}} an [[unbiased estimator]] of <math>\lambda,</math> although <math>\overline{x}</math> {{em|is}} an unbiased<ref name="Dean W. Wichern-2007">{{cite book|author1=Richard Arnold Johnson|author2=Dean W. Wichern|title=Applied Multivariate Statistical Analysis|url=https://books.google.com/books?id=gFWcQgAACAAJ|access-date=10 August 2012|year=2007 |publisher=Pearson Prentice Hall|isbn=978-0-13-187715-3}}</ref> MLE<ref>''[http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm NIST/SEMATECH e-Handbook of Statistical Methods]''</ref> estimator of <math>1/\lambda</math> and the distribution mean.

The bias of <math> \widehat{\lambda}_\text{mle} </math> is equal to
<math display="block">B \equiv \operatorname{E}\left[\left(\widehat{\lambda}_\text{mle} - \lambda\right)\right] = \frac{\lambda}{n - 1} </math>
which yields the [[Maximum likelihood estimation#Second-order efficiency after correction for bias|bias-corrected maximum likelihood estimator]]
<math display="block">\widehat{\lambda}^*_\text{mle} = \widehat{\lambda}_\text{mle} - B.</math>

An approximate minimizer of [[mean squared error]] (see also: [[bias–variance tradeoff]]) can be found, assuming a sample size greater than two, with a correction factor to the MLE:
<math display="block">\widehat{\lambda} = \left(\frac{n - 2}{n}\right) \left(\frac{1}{\bar{x}}\right) = \frac{n - 2}{\sum_i x_i}</math>
This is derived from the mean and variance of the [[inverse-gamma distribution]], <math display="inline">\mbox{Inv-Gamma}(n, \lambda)</math>.<ref>{{cite journal |first1=Abdulaziz |last1=Elfessi |first2=David M. |last2=Reineke |title=A Bayesian Look at Classical Estimation: The Exponential Distribution |journal=Journal of Statistics Education |volume=9 |issue=1 |year=2001 |doi=10.1080/10691898.2001.11910648|doi-access=free }}</ref>

===Fisher information===
The [[Fisher information]], denoted <math>\mathcal{I}(\lambda)</math>, for an estimator of the rate parameter <math>\lambda</math> is given as:
<math display="block">\mathcal{I}(\lambda) = \operatorname{E} \left[\left. \left(\frac{\partial}{\partial\lambda} \log f(x;\lambda)\right)^2\right|\lambda\right] = \int \left(\frac{\partial}{\partial\lambda} \log f(x;\lambda)\right)^2 f(x; \lambda)\,dx</math>

Plugging in the distribution and solving gives:
<math display="block"> \mathcal{I}(\lambda) = \int_{0}^{\infty} \left(\frac{\partial}{\partial\lambda} \log \lambda e^{-\lambda x}\right)^2 \lambda e^{-\lambda x}\,dx = \int_{0}^{\infty} \left(\frac{1}{\lambda} - x\right)^2 \lambda e^{-\lambda x}\,dx = \lambda^{-2}.</math>

This determines the amount of information each independent sample of an exponential distribution carries about the unknown rate parameter <math>\lambda</math>.

===Confidence intervals===
An exact 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=https://books.google.com/books?id=mXP_UEiUo9wC&pg=PA267| edition=4th|year=2009| publisher=Associated Press| isbn=978-0-12-370483-2}}</ref>
<math display="block">\frac{2n}{\widehat{\lambda}_{\textrm{mle}} \chi^2_{\frac{\alpha}{2},2n} }< \frac{1}{\lambda} < \frac{2n}{\widehat{\lambda}_{\textrm{mle}}  \chi^2_{1-\frac{\alpha}{2},2n}}\,,</math>
which is also equal to
<math display="block">\frac{2n\overline{x}}{\chi^2_{\frac{\alpha}{2},2n}} < \frac{1}{\lambda} < \frac{2n\overline{x}}{\chi^2_{1-\frac{\alpha}{2},2n}}\,,</math>
where {{math|χ{{su|p=2|b=''p'',''v''}}}} is the {{math|100(''p'')}} [[percentile]] of the  [[chi squared distribution]] with ''v'' [[degrees of freedom (statistics)|degrees of freedom]], n is the number of observations 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''{{su|p=2|b=''p'',''v''}}}} distribution. This approximation gives the following values for a 95% confidence interval:
<math display="block">\begin{align}
  \lambda_\text{lower} &= \widehat{\lambda}\left(1 - \frac{1.96}{\sqrt{n}}\right) \\
  \lambda_\text{upper} &= \widehat{\lambda}\left(1 + \frac{1.96}{\sqrt{n}}\right)
\end{align}</math>

This approximation may be acceptable for samples containing at least 15 to 20 elements.<ref name="Guerriero-2012">{{Cite journal | first1 = V. | last1= Guerriero | year = 2012  | title = Power Law Distribution: Method of Multi-scale Inferential Statistics| journal = Journal of Modern Mathematics Frontier | url =https://www.academia.edu/27459041 | volume = 1  | pages = 21–28}}</ref>

===Bayesian inference===
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:

<math display="block">\operatorname{Gamma}(\lambda; \alpha, \beta) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda\beta).</math>

The [[posterior distribution]] ''p'' can then be expressed in terms of the likelihood function defined above and a gamma prior:

<math display="block">\begin{align}
  p(\lambda) &\propto L(\lambda) \Gamma(\lambda; \alpha, \beta) \\
    &= \lambda^n \exp\left(-\lambda n\overline{x}\right) \frac{\beta^{\alpha}}{\Gamma(\alpha)} \lambda^{\alpha-1} \exp(-\lambda \beta) \\
    &\propto \lambda^{(\alpha+n)-1} \exp(-\lambda \left(\beta + n\overline{x}\right)).
\end{align}</math>

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:

<math display="block">p(\lambda) = \operatorname{Gamma}(\lambda; \alpha + n, \beta + n\overline{x}).</math>

Here the [[Hyperparameter (Bayesian statistics)|hyperparameter]] ''α'' can be interpreted as the number of prior observations, and ''β'' as the sum of the prior observations.
The posterior mean here is:
<math display="block">\frac{\alpha + n}{\beta + n\overline{x}}.</math>

==Occurrence and applications==

===Occurrence of events===
The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneous [[Poisson process]].

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.

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:
* The time until a radioactive [[particle decay]]s, or the time between clicks of a [[Geiger counter]]
* The time between receiving one telephone call and the next
* The time until default (on payment to company debt holders) in reduced-form credit risk modeling

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.

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 arrival of customers for instance is also modeled by the [[Poisson distribution]] if the arrivals are independent and distributed identically.)  The length of a process that can be thought of as a sequence of several independent tasks follows the [[Erlang distribution]] (which is the distribution of the sum of several independent exponentially distributed variables).
[[Reliability theory]] and [[reliability engineering]] also make extensive use of the exponential distribution. Because of the 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.
[[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>]]

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.

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|editor-last=Ritzema|editor-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=[https://archive.org/details/drainageprincipl0000unse/page/175 175–224]|url=https://archive.org/details/drainageprincipl0000unse/page/175| isbn=90-70754-33-9}}</ref>

: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]].
In operating-rooms management, the distribution of surgery duration for a category of surgeries with [[Predictive methods for surgery duration|no typical work-content]] (like in an emergency room, encompassing all types of surgeries).

===Prediction===
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
<math display="block">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>

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.

A predictive distribution free of the issues of choosing priors that arise under the subjective Bayesian approach is

<math display="block">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>

which can be considered as
# a frequentist [[confidence distribution]], obtained from the distribution of the pivotal quantity <math>{x_{n+1}}/{\overline{x}}</math>;<ref>{{cite journal |last1=Lawless |first1=J. F. |last2=Fredette |first2=M. |title=Frequentist predictions intervals and predictive distributions |journal=Biometrika |year=2005 |volume=92 |issue=3 |pages=529–542 |doi=10.1093/biomet/92.3.529 |doi-access= }}</ref>
# 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>{{cite journal | last1 = Bjornstad | first1 = J.F. | year = 1990 | title = Predictive Likelihood: A Review | journal = Statist. Sci. | volume = 5 | issue = 2| pages = 242–254 | doi=10.1214/ss/1177012175| doi-access = free }}</ref>
# an objective Bayesian predictive posterior distribution, obtained using the non-informative [[Jeffreys prior]] 1/''λ'';
# 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>

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

<math display="block">\begin{align}
\operatorname{E}_{\lambda_0} \left[ \Delta(\lambda_0\parallel p_{\rm ML}) \right] &= \psi(n) + \frac{1}{n-1} - \log(n) \\
\operatorname{E}_{\lambda_0} \left[ \Delta(\lambda_0\parallel p_{\rm CNML}) \right] &= \psi(n) + \frac{1}{n} - \log(n)
\end{align}</math>

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}}.

==Random variate generation==
{{further|Non-uniform random variate generation}}

<!-- This section is linked from [[Gamma distribution]] -->
A conceptually very simple method for generating exponential [[variate]]s is based on [[inverse transform sampling]]: Given a random variate ''U'' drawn from the [[uniform distribution (continuous)|uniform distribution]] on the unit interval {{open-open|0, 1}}, the variate

<math display="block">T = F^{-1}(U)</math>

has an exponential distribution, where ''F''{{i sup|−1}} is the [[quantile function]], defined by

<math display="block">F^{-1}(p)=\frac{-\ln(1-p)}{\lambda}.</math>

Moreover, if ''U'' is uniform on (0, 1), then so is 1 − ''U''.  This means one can generate exponential variates as follows:

<math display="block">T = \frac{-\ln(U)}{\lambda}.</math>

Other methods for generating exponential variates are discussed by Knuth<ref>[[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="Luc 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>

A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.<ref name="Luc Devroye"/>

==See also==
* [[Dead time]] – an application of exponential distribution to particle detector analysis.
* [[Laplace distribution]], or the "double exponential distribution".
* [[Relationships among probability distributions]]
* [[Marshall–Olkin exponential distribution]]

==References==
{{Reflist}}

==External links==
{{Wikibooks|Probability|Important Distributions#Exponential distribution|Exponential distribution}}
* {{springer|title=Exponential distribution|id=p/e036900}}
* [http://www.elektro-energetika.cz/calculations/ex.php?language=english Online calculator of Exponential Distribution]

{{ProbDistributions|continuous-semi-infinite}}
{{Authority control}}

{{DEFAULTSORT:Exponential Distribution}}
[[Category:Continuous distributions]]
[[Category:Exponentials]]
[[Category:Poisson point processes]]
[[Category:Conjugate prior distributions]]
[[Category:Exponential family distributions]]
[[Category:Infinitely divisible probability distributions]]
[[Category:Survival analysis]]