Movatterモバイル変換

[0]ホーム

Jump to content

Exponential distribution

Edit links

From Wikipedia, the free encyclopedia

Probability distribution

Not to be confused with theexponential family of probability distributions.

Exponential
Probability density function
Cumulative distribution function
Parameters	$\lambda >0,$ rate, or inversescale
Support	$x\in [0,\infty )$
PDF	$\lambda e^{-\lambda x}$
CDF	$1-e^{-\lambda x}$
Quantile	$-{\frac {\ln(1-p)}{\lambda }}$
Mean	${\frac {1}{\lambda }}$
Median	${\frac {\ln 2}{\lambda }}$
Mode	$0 {\displaystyle 0}$
Variance	${\frac {1}{\lambda ^{2}}}$
Skewness	$2 {\displaystyle 2}$
Excess kurtosis	$6 {\displaystyle 6}$
Entropy	$1-\ln \lambda$
MGF	${\frac {\lambda }{\lambda -t}},{\text{ for }}t<\lambda$
CF	${\frac {\lambda }{\lambda -it}}$
Fisher information	${\frac {1}{\lambda ^{2}}}$
Kullback–Leibler divergence	$\ln {\frac {\lambda _{0}}{\lambda }}+{\frac {\lambda }{\lambda _{0}}}-1$
Expected shortfall	${\frac {-\ln(1-p)+1}{\lambda }}$

Inprobability theory andstatistics, theexponential distribution ornegative exponential distribution is theprobability distribution of the distance between events in aPoisson 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.^[1] It is a particular case of thegamma distribution. It is the continuous analogue of thegeometric distribution, and it has the key property of beingmemoryless.^[2] In addition to being used for the analysis of Poisson point processes it is found in various other contexts.^[3]

The exponential distribution is not the same as the class ofexponential 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, such as thenormal,binomial,gamma, andPoisson distributions.^[3]

Definitions

Probability density function

Theprobability density function (pdf) of an exponential distribution is

f(x;\lambda )={\begin{cases}\lambda e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}

Hereλ > 0 is the parameter of the distribution, often called therate parameter. The distribution is supported on the interval [0, ∞). If arandom variableX has this distribution, we write X ~ Exp(λ).

The exponential distribution exhibitsinfinite divisibility.

Cumulative distribution function

Thecumulative distribution function is given by

F(x;\lambda )={\begin{cases}1-e^{-\lambda x}&x\geq 0,\\0&x<0.\end{cases}}

Alternative parametrization

The exponential distribution is sometimes parametrized in terms of thescale parameterβ = 1/λ, which is also the mean: $f(x;\beta )={\begin{cases}{\frac {1}{\beta }}e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}\qquad \qquad F(x;\beta )={\begin{cases}1-e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}$

Properties

Mean, variance, moments, and median

The mean is the probability mass centre, that is, thefirst moment.

The mean orexpected value of an exponentially distributed random variableX with rate parameterλ is given by $\operatorname {E} [X]={\frac {1}{\lambda }}.$

In light of the examples givenbelow, 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.

Thevariance ofX is given by $\operatorname {Var} [X]={\frac {1}{\lambda ^{2}}},$ so thestandard deviation is equal to the mean.

Themoments ofX, for $n\in \mathbb {N}$ are given by $\operatorname {E} \left[X^{n}\right]={\frac {n!}{\lambda ^{n}}}.$

Thecentral moments ofX, for $n\in \mathbb {N}$ are given by $\mu _{n}={\frac {!n}{\lambda ^{n}}}={\frac {n!}{\lambda ^{n}}}\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.$ where !n is thesubfactorial ofn

Themedian ofX is given by $\operatorname {m} [X]={\frac {\ln(2)}{\lambda }}<\operatorname {E} [X],$ whereln refers to thenatural logarithm. Thus theabsolute difference between the mean and median is $\left|\operatorname {E} \left[X\right]-\operatorname {m} \left[X\right]\right|={\frac {1-\ln(2)}{\lambda }}<{\frac {1}{\lambda }}=\operatorname {\sigma } [X],$

in accordance with themedian-mean inequality.

Memorylessness property of exponential random variable

An exponentially distributed random variableT obeys the relation $\Pr \left(T>s+t\mid T>s\right)=\Pr(T>t),\qquad \forall s,t\geq 0.$

This can be seen by considering thecomplementary cumulative distribution function: ${\begin{aligned}\Pr \left(T>s+t\mid T>s\right)&={\frac {\Pr \left(T>s+t\cap T>s\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {\Pr \left(T>s+t\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {e^{-\lambda (s+t)}}{e^{-\lambda s}}}\\[4pt]&=e^{-\lambda t}\\[4pt]&=\Pr(T>t).\end{aligned}}$

WhenT is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, ifT is conditioned on a failure to observe the event over some initial period of times, 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, theconditional 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 thegeometric distribution arethe only memoryless probability distributions.

The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constantfailure rate.

Quantiles

Tukey anomaly criteria for exponential probability distribution function. — Tukey criteria for anomalies.^{[citation needed]}

Thequantile function (inverse cumulative distribution function) for Exp(λ) is $F^{-1}(p;\lambda )={\frac {-\ln(1-p)}{\lambda }},\qquad 0\leq p<1$

Thequartiles are therefore:

first quartile: ln(4/3)/λ
median: ln(2)/λ
third quartile: ln(4)/λ

And as a consequence theinterquartile range is ln(3)/λ.

Conditional Value at Risk (Expected Shortfall)

The conditional value at risk (CVaR) also known as theexpected shortfall or superquantile for Exp(λ) is derived as follows:^[4]

${\begin{aligned}{\bar {q}}_{\alpha }(X)&={\frac {1}{1-\alpha }}\int _{\alpha }^{1}q_{p}(X)dp\\&={\frac {1}{(1-\alpha )}}\int _{\alpha }^{1}{\frac {-\ln(1-p)}{\lambda }}dp\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{1-\alpha }^{0}-\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{0}^{1-\alpha }\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}[(1-\alpha )\ln(1-\alpha )-(1-\alpha )]\\&={\frac {-\ln(1-\alpha )+1}{\lambda }}\\\end{aligned}}$

Buffered Probability of Exceedance (bPOE)

Main article:Buffered probability of exceedance

The buffered probability of exceedance is one minus the probability level at which the CVaR equals the threshold $x {\displaystyle x}$ . It is derived as follows:^[4]

${\begin{aligned}{\bar {p}}_{x}(X)&=\{1-\alpha |{\bar {q}}_{\alpha }(X)=x\}\\&=\{1-\alpha |{\frac {-\ln(1-\alpha )+1}{\lambda }}=x\}\\&=\{1-\alpha |\ln(1-\alpha )=1-\lambda x\}\\&=\{1-\alpha |e^{\ln(1-\alpha )}=e^{1-\lambda x}\}=\{1-\alpha |1-\alpha =e^{1-\lambda x}\}=e^{1-\lambda x}\end{aligned}}$

Kullback–Leibler divergence

The directedKullback–Leibler divergence innats of $e^{\lambda }$ ("approximating" distribution) from $e^{\lambda _{0}}$ ('true' distribution) is given by ${\begin{aligned}\Delta (\lambda _{0}\parallel \lambda )&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {p_{\lambda _{0}}(x)}{p_{\lambda }(x)}}\right)\\&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {\lambda _{0}e^{\lambda _{0}x}}{\lambda e^{\lambda x}}}\right)\\&=\log(\lambda _{0})-\log(\lambda )-(\lambda _{0}-\lambda )E_{\lambda _{0}}(x)\\&=\log(\lambda _{0})-\log(\lambda )+{\frac {\lambda }{\lambda _{0}}}-1.\end{aligned}}$

Maximum entropy distribution

Among all continuous probability distributions withsupport[0, ∞) and meanμ, the exponential distribution withλ = 1/μ has the largestdifferential entropy. In other words, it is themaximum entropy probability distribution for arandom variateX which is greater than or equal to zero and for which E[X] is fixed.^[5]

Distribution of the minimum of exponential random variables

LetX₁, ...,X_n beindependent exponentially distributed random variables with rate parametersλ₁, ...,λ_n. Then $\min \left\{X_{1},\dotsc ,X_{n}\right\}$ is also exponentially distributed, with parameter $\lambda =\lambda _{1}+\dotsb +\lambda _{n}.$

This can be seen by considering thecomplementary cumulative distribution function: ${\begin{aligned}&\Pr \left(\min\{X_{1},\dotsc ,X_{n}\}>x\right)\\={}&\Pr \left(X_{1}>x,\dotsc ,X_{n}>x\right)\\={}&\prod _{i=1}^{n}\Pr \left(X_{i}>x\right)\\={}&\prod _{i=1}^{n}\exp \left(-x\lambda _{i}\right)=\exp \left(-x\sum _{i=1}^{n}\lambda _{i}\right).\end{aligned}}$

The index of the variable which achieves the minimum is distributed according to the categorical distribution $\Pr \left(X_{k}=\min\{X_{1},\dotsc ,X_{n}\}\right)={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.$

A proof can be seen by letting $I=\operatorname {argmin} _{i\in \{1,\dotsb ,n\}}\{X_{1},\dotsc ,X_{n}\}$ . Then, ${\begin{aligned}\Pr(I=k)&=\int _{0}^{\infty }\Pr(X_{k}=x)\Pr(\forall _{i\neq k}X_{i}>x)\,dx\\&=\int _{0}^{\infty }\lambda _{k}e^{-\lambda _{k}x}\left(\prod _{i=1,i\neq k}^{n}e^{-\lambda _{i}x}\right)dx\\&=\lambda _{k}\int _{0}^{\infty }e^{-\left(\lambda _{1}+\dotsb +\lambda _{n}\right)x}dx\\&={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.\end{aligned}}$

Note that $\max\{X_{1},\dotsc ,X_{n}\}$ is not exponentially distributed, ifX₁, ...,X_n do not all have parameter 0.^[6]

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

Let $X_{1},\dotsc ,X_{n}$ be $n {\displaystyle n}$ independent and identically distributed exponential random variables with rate parameterλ.Let $X_{(1)},\dotsc ,X_{(n)}$ denote the correspondingorder statistics.For $i<j$ , the joint moment $\operatorname {E} \left[X_{(i)}X_{(j)}\right]$ of the order statistics $X_{(i)}$ and $X_{(j)}$ is given by ${\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right]\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}+\sum _{k=0}^{i-1}{\frac {1}{((n-k)\lambda )^{2}}}+\left(\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}\right)^{2}.\end{aligned}}$

This can be seen by invoking thelaw of total expectation and the memoryless property: ${\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\int _{0}^{\infty }\operatorname {E} \left[X_{(i)}X_{(j)}\mid X_{(i)}=x\right]f_{X_{(i)}}(x)\,dx\\&=\int _{x=0}^{\infty }x\operatorname {E} \left[X_{(j)}\mid X_{(j)}\geq x\right]f_{X_{(i)}}(x)\,dx&&\left({\textrm {since}}~X_{(i)}=x\implies X_{(j)}\geq x\right)\\&=\int _{x=0}^{\infty }x\left[\operatorname {E} \left[X_{(j)}\right]+x\right]f_{X_{(i)}}(x)\,dx&&\left({\text{by the memoryless property}}\right)\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right].\end{aligned}}$

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

Sum of two independent exponential random variables

The probability distribution function (PDF) of a sum of two independent random variables is theconvolution of their individual PDFs. If $X_{1}$ and $X_{2}$ are independent exponential random variables with respective rate parameters $\lambda _{1}$ and $\lambda _{2},$ then the probability density of $Z=X_{1}+X_{2}$ is given by ${\begin{aligned}f_{Z}(z)&=\int _{-\infty }^{\infty }f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\,dx_{1}\\&=\int _{0}^{z}\lambda _{1}e^{-\lambda _{1}x_{1}}\lambda _{2}e^{-\lambda _{2}(z-x_{1})}\,dx_{1}\\&=\lambda _{1}\lambda _{2}e^{-\lambda _{2}z}\int _{0}^{z}e^{(\lambda _{2}-\lambda _{1})x_{1}}\,dx_{1}\\&={\begin{cases}{\dfrac {\lambda _{1}\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\left(e^{-\lambda _{1}z}-e^{-\lambda _{2}z}\right)&{\text{ if }}\lambda _{1}\neq \lambda _{2}\\[4pt]\lambda ^{2}ze^{-\lambda z}&{\text{ if }}\lambda _{1}=\lambda _{2}=\lambda .\end{cases}}\end{aligned}}$ The entropy of this distribution is available in closed form: assuming $\lambda _{1}>\lambda _{2}$ (without loss of generality), then ${\begin{aligned}H(Z)&=1+\gamma +\ln \left({\frac {\lambda _{1}-\lambda _{2}}{\lambda _{1}\lambda _{2}}}\right)+\psi \left({\frac {\lambda _{1}}{\lambda _{1}-\lambda _{2}}}\right),\end{aligned}}$ where $\gamma$ is theEuler-Mascheroni constant, and $\psi (\cdot )$ is thedigamma function.^[7]

In the case of equal rate parameters, the result is anErlang distribution with shape 2 and parameter $\lambda ,$ which in turn is a special case ofgamma distribution.

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

Related distributions

IfX ~Laplace(μ, β⁻¹), then |X − μ| ~ Exp(β).^[8]
IfX ~U(0, 1) then −log(X) ~ Exp(1).
IfX ~Pareto(1, λ), then log(X) ~ Exp(λ).^[8]
IfX ~SkewLogistic(θ), then $\log \left(1+e^{-X}\right)\sim \operatorname {Exp} (\theta )$ .
IfX_i ~U(0, 1) then $\lim _{n\to \infty }n\min \left(X_{1},\ldots ,X_{n}\right)\sim \operatorname {Exp} (1)$
The exponential distribution is a limit of a scaledbeta distribution: $\lim _{n\to \infty }n\operatorname {Beta} (1,n)=\operatorname {Exp} (1).$
The exponential distribution is a special case of type 3Pearson distribution.
The exponential distribution is the special case of aGamma distribution with shape parameter 1.^[8]
IfX ~ Exp(λ) andX_i ~ Exp(λ_i) then:
- $kX\sim \operatorname {Exp} \left({\frac {\lambda }{k}}\right)$ , closure under scaling by a positive factor.
- 1 + X ~BenktanderWeibull(λ, 1), which reduces to a truncated exponential distribution.
- ke^X ~Pareto(k, λ).^[8]
- e^−λX ~U(0, 1).
- e^−X ~Beta(λ, 1).^[8]
- ⁠1/k⁠e^X ~PowerLaw(k, λ)
- ${\sqrt {X}}\sim \operatorname {Rayleigh} \left({\frac {1}{\sqrt {2\lambda }}}\right)$ , theRayleigh distribution^[8]
- $X\sim \operatorname {Weibull} \left({\frac {1}{\lambda }},1\right)$ , theWeibull distribution^[8]
- $X^{2}\sim \operatorname {Weibull} \left({\frac {1}{\lambda ^{2}}},{\frac {1}{2}}\right)$ ^[8]
- μ − β log(λX) ∼Gumbel(μ, β).
- $\lfloor X\rfloor \sim \operatorname {Geometric} \left(1-e^{-\lambda }\right)$ , ageometric distribution on 0,1,2,3,...
- $\lceil X\rceil \sim \operatorname {Geometric} \left(1-e^{-\lambda }\right)$ , ageometric distribution on 1,2,3,4,...
- If alsoY ~ Erlang(n, λ) or $Y\sim \Gamma \left(n,{\frac {1}{\lambda }}\right)$ then ${\frac {X}{Y}}+1\sim \operatorname {Pareto} (1,n)$
- If also λ ~Gamma(k, θ) (shape, scale parametrisation) then the marginal distribution ofX isLomax(k, 1/θ), the gammamixture
- λ₁X₁ − λ₂Y₂ ~Laplace(0, 1).
- min{X₁, ...,X_n} ~ Exp(λ₁ + ... + λ_n).
- If also λ_i = λ then:
  - $X_{1}+\cdots +X_{k}=\sum _{i}X_{i}\sim$ Erlang(k, λ) =Gamma(k, λ) with integer shape parameterk and rate parameter λ.^[9]
  - If $T=(X_{1}+\cdots +X_{n})=\sum _{i=1}^{n}X_{i}$ , then $2\lambda T\sim \chi _{2n}^{2}$ .
  - X_i −X_j ~ Laplace(0, λ⁻¹).
- If alsoX_i are independent, then:
  - ${\frac {X_{i}}{X_{i}+X_{j}}}$ ~U(0, 1)
  - $Z={\frac {\lambda _{i}X_{i}}{\lambda _{j}X_{j}}}$ has probability density function $f_{Z}(z)={\frac {1}{(z+1)^{2}}}$ . This can be used to obtain aconfidence interval for ${\frac {\lambda _{i}}{\lambda _{j}}}$ .
- If also λ = 1:
  - $\mu -\beta \log \left({\frac {e^{-X}}{1-e^{-X}}}\right)\sim \operatorname {Logistic} (\mu ,\beta )$ , thelogistic distribution
  - $\mu -\beta \log \left({\frac {X_{i}}{X_{j}}}\right)\sim \operatorname {Logistic} (\mu ,\beta )$
  - μ − σ log(X) ~GEV(μ, σ, 0).
  - Further if $Y\sim \Gamma \left(\alpha ,{\frac {\beta }{\alpha }}\right)$ then ${\sqrt {XY}}\sim \operatorname {K} (\alpha ,\beta )$ (K-distribution)
- If also λ = 1/2 thenX ∼ χ²
  ₂; i.e.,X has achi-squared distribution with 2degrees of freedom. Hence: $\operatorname {Exp} (\lambda )={\frac {1}{2\lambda }}\operatorname {Exp} \left({\frac {1}{2}}\right)\sim {\frac {1}{2\lambda }}\chi _{2}^{2}\Rightarrow \sum _{i=1}^{n}\operatorname {Exp} (\lambda )\sim {\frac {1}{2\lambda }}\chi _{2n}^{2}$
If $X\sim \operatorname {Exp} \left({\frac {1}{\lambda }}\right)$ and $Y\mid X$ ~Poisson(X) then $Y\sim \operatorname {Geometric} \left({\frac {1}{1+\lambda }}\right)$ (geometric distribution)
TheHoyt distribution can be obtained from exponential distribution andarcsine distribution
The exponential distribution is a limit of theκ-exponential distribution in the $\kappa =0$ case.
Exponential distribution is a limit of theκ-Generalized Gamma distribution in the $\alpha =1$ and $\nu =1$ cases:
$\lim _{(\alpha ,\nu )\to (0,1)}p_{\kappa }(x)=(1+\kappa \nu )(2\kappa )^{\nu }{\frac {\Gamma {\Big (}{\frac {1}{2\kappa }}+{\frac {\nu }{2}}{\Big )}}{\Gamma {\Big (}{\frac {1}{2\kappa }}-{\frac {\nu }{2}}{\Big )}}}{\frac {\alpha \lambda ^{\nu }}{\Gamma (\nu )}}x^{\alpha \nu -1}\exp _{\kappa }(-\lambda x^{\alpha })=\lambda e^{-\lambda x}$

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.^[8]
exGaussian distribution – the sum of an exponential distribution and anormal distribution.

Statistical inference

Below, suppose random variableX is exponentially distributed with rate parameter λ, and $x_{1},\dotsc ,x_{n}$ aren independent samples fromX, with sample mean ${\bar {x}}$ .

Parameter estimation

Themaximum likelihood estimator for λ is constructed as follows.

Thelikelihood function for λ, given anindependent and identically distributed samplex = (x₁, ...,x_n) drawn from the variable, is: $L(\lambda )=\prod _{i=1}^{n}\lambda \exp(-\lambda x_{i})=\lambda ^{n}\exp \left(-\lambda \sum _{i=1}^{n}x_{i}\right)=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right),$

where: ${\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}$ is the sample mean.

The derivative of the likelihood function's logarithm is: ${\frac {d}{d\lambda }}\ln L(\lambda )={\frac {d}{d\lambda }}\left(n\ln \lambda -\lambda n{\overline {x}}\right)={\frac {n}{\lambda }}-n{\overline {x}}\ {\begin{cases}>0,&0<\lambda <{\frac {1}{\overline {x}}},\\[8pt]=0,&\lambda ={\frac {1}{\overline {x}}},\\[8pt]<0,&\lambda >{\frac {1}{\overline {x}}}.\end{cases}}$

Consequently, themaximum likelihood estimate for the rate parameter is: ${\widehat {\lambda }}_{\text{mle}}={\frac {1}{\overline {x}}}={\frac {n}{\sum _{i}x_{i}}}$

This isnot anunbiased estimator of $\lambda ,$ although ${\overline {x}}$ is an unbiased^[10] MLE^[11] estimator of $1/\lambda$ and the distribution mean.

The bias of ${\widehat {\lambda }}_{\text{mle}}$ is equal to $B\equiv \operatorname {E} \left[\left({\widehat {\lambda }}_{\text{mle}}-\lambda \right)\right]={\frac {\lambda }{n-1}}$ which yields thebias-corrected maximum likelihood estimator ${\widehat {\lambda }}_{\text{mle}}^{*}={\widehat {\lambda }}_{\text{mle}}-B.$

An approximate minimizer ofmean squared error (see also:bias–variance tradeoff) can be found, assuming a sample size greater than two, with a correction factor to the MLE: ${\widehat {\lambda }}=\left({\frac {n-2}{n}}\right)\left({\frac {1}{\bar {x}}}\right)={\frac {n-2}{\sum _{i}x_{i}}}$ This is derived from the mean and variance of theinverse-gamma distribution, ${\textstyle {\mbox{Inv-Gamma}}(n,\lambda )}$ .^[12]

Fisher information

TheFisher information, denoted ${\mathcal {I}}(\lambda )$ , for an estimator of the rate parameter $\lambda$ is given as: ${\mathcal {I}}(\lambda )=\operatorname {E} \left[\left.\left({\frac {\partial }{\partial \lambda }}\log f(x;\lambda )\right)^{2}\right|\lambda \right]=\int \left({\frac {\partial }{\partial \lambda }}\log f(x;\lambda )\right)^{2}f(x;\lambda )\,dx$

Plugging in the distribution and solving gives: ${\mathcal {I}}(\lambda )=\int _{0}^{\infty }\left({\frac {\partial }{\partial \lambda }}\log \lambda e^{-\lambda x}\right)^{2}\lambda e^{-\lambda x}\,dx=\int _{0}^{\infty }\left({\frac {1}{\lambda }}-x\right)^{2}\lambda e^{-\lambda x}\,dx=\lambda ^{-2}.$

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

Confidence intervals

An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by:^[13] ${\frac {2n}{{\widehat {\lambda }}_{\textrm {mle}}\chi _{{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n}{{\widehat {\lambda }}_{\textrm {mle}}\chi _{1-{\frac {\alpha }{2}},2n}^{2}}}\,,$ which is also equal to ${\frac {2n{\overline {x}}}{\chi _{{\frac {\alpha }{2}},2n}^{2}}}<{\frac {1}{\lambda }}<{\frac {2n{\overline {x}}}{\chi _{1-{\frac {\alpha }{2}},2n}^{2}}}\,,$ whereχ²
_p,v is the100(p)percentile of thechi squared distribution withvdegrees 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 theχ²
_p,v distribution. This approximation gives the following values for a 95% confidence interval: ${\begin{aligned}\lambda _{\text{lower}}&={\widehat {\lambda }}\left(1-{\frac {1.96}{\sqrt {n}}}\right)\\\lambda _{\text{upper}}&={\widehat {\lambda }}\left(1+{\frac {1.96}{\sqrt {n}}}\right)\end{aligned}}$

This approximation may be acceptable for samples containing at least 15 to 20 elements.^[14]

Bayesian inference with a conjugate prior

Theconjugate prior for the exponential distribution is thegamma distribution (of which the exponential distribution is a special case). The following parameterization of the gamma probability density function is useful:

$\operatorname {Gamma} (\lambda ;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\lambda ^{\alpha -1}\exp(-\lambda \beta ).$

Theposterior distributionp can then be expressed in terms of the likelihood function defined above and a gamma prior:

${\begin{aligned}p(\lambda )&\propto L(\lambda )\Gamma (\lambda ;\alpha ,\beta )\\&=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right){\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\lambda ^{\alpha -1}\exp(-\lambda \beta )\\&\propto \lambda ^{(\alpha +n)-1}\exp(-\lambda \left(\beta +n{\overline {x}}\right)).\end{aligned}}$

Now the posterior densityp 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:

$p(\lambda )=\operatorname {Gamma} (\lambda ;\alpha +n,\beta +n{\overline {x}}).$

Here thehyperparameterα can be interpreted as the number of prior observations, andβ as the sum of the prior observations.The posterior mean here is: ${\frac {\alpha +n}{\beta +n{\overline {x}}}}.$

Bayesian inference with a calibrating prior

The exponential distribution is one of a number of statistical distributions withgroup structure. As a result of the group structure, the exponential has an associatedHaar measure, which is $1/\lambda .$ The use of theHaar measure as theprior (known as the Haar prior) in a Bayesian prediction gives probabilities that are perfectly calibrated, for any underlying true parameter values.^[15]^[16]^[17] Perfectly calibrated probabilities have the property that the predicted probabilities match the frequency of out-of-sample events exactly. For the exponential, there is an exact expression for Bayesian predictions generated using the Haar prior, given by

$p_{\rm {Haar-prior}}(x_{n+1}\mid x_{1},\ldots ,x_{n})={\frac {n^{n+1}\left({\overline {x}}\right)^{n}}{\left(n{\overline {x}}+x_{n+1}\right)^{n+1}}}.$

This is an example of calibrating prior prediction, in which the prior is chosen to improve calibration (and, in this case, to make the calibration perfect). Calibrating prior prediction for the exponential using the Haar prior is implemented in theR software package fitdistcp.[1]

The same prediction can be derived from a number of other perspectives, as discussed in theprediction section below.

Occurrence and applications

Occurrence of events

The exponential distribution occurs naturally when describing the lengths of the inter-arrival times in a homogeneousPoisson process.

The exponential distribution may be viewed as a continuous counterpart of thegeometric distribution, which describes the number ofBernoulli trials necessary for adiscrete 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 radioactiveparticle decays, or the time between clicks of aGeiger 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 betweenmutations on aDNA strand, or betweenroadkills on a given road.

Inqueuing 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 thePoisson 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 theErlang distribution (which is the distribution of the sum of several independent exponentially distributed variables).

Reliability theory andreliability engineering also make extensive use of the exponential distribution. Because of the memoryless property of this distribution, it is well-suited to model the constanthazard rate portion of thebathtub curve used in reliability theory. It is also very convenient because it is so easy to addfailure rates 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.

Fitted cumulative exponential distribution to annually maximum 1-day rainfalls

Inphysics, if you observe agas at a fixedtemperature andpressure in a uniformgravitational field, the heights of the various molecules also follow an approximate exponential distribution, known as theBarometric formula. This is a consequence of the entropy property mentioned below.

Inhydrology, 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.^[18]

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 thebinomial distribution. The rainfall data are represented byplotting positions as part of thecumulative frequency analysis.

In operating-rooms management, the distribution of surgery duration for a category of surgeries withno typical work-content (like in an emergency room, encompassing all types of surgeries).

Prediction

Having observed a sample ofn 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 parameterλ 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 samplex_n+1, conditioned on the observed samplesx = (x₁, ...,x_n) given by $p_{\rm {ML}}(x_{n+1}\mid x_{1},\ldots ,x_{n})=\left({\frac {1}{\overline {x}}}\right)\exp \left(-{\frac {x_{n+1}}{\overline {x}}}\right).$

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

$p_{\rm {CNML}}(x_{n+1}\mid x_{1},\ldots ,x_{n})={\frac {n^{n+1}\left({\overline {x}}\right)^{n}}{\left(n{\overline {x}}+x_{n+1}\right)^{n+1}}},$

which can be considered as

a frequentistconfidence distribution, obtained from the distribution of the pivotal quantity ${x_{n+1}}/{\overline {x}}$ ;^[19]
a profile predictive likelihood, obtained by eliminating the parameterλ from the joint likelihood ofx_n+1 andλ by maximization;^[20]
an objective Bayesian predictive posterior distribution, obtained using the non-informativeJeffreys prior 1/λ, which is equal to the right Haar prior in this case. Predictions generated using the right Haar prior are guaranteed to give perfectly calibrated probabilities.^[21]^[22]
the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.^[23]

The accuracy of a predictive distribution may be measured using the distance or divergence between the true exponential distribution with rate parameter,λ₀, and the predictive distribution based on the samplex. TheKullback–Leibler divergence is a commonly used, parameterisation free measure of the difference between two distributions. Letting Δ(λ₀||p) denote the Kullback–Leibler divergence between an exponential with rate parameterλ₀ and a predictive distributionp it can be shown that

${\begin{aligned}\operatorname {E} _{\lambda _{0}}\left[\Delta (\lambda _{0}\parallel p_{\rm {ML}})\right]&=\psi (n)+{\frac {1}{n-1}}-\log(n)\\\operatorname {E} _{\lambda _{0}}\left[\Delta (\lambda _{0}\parallel p_{\rm {CNML}})\right]&=\psi (n)+{\frac {1}{n}}-\log(n)\end{aligned}}$

where the expectation is taken with respect to the exponential distribution with rate parameterλ₀ ∈ (0, ∞), andψ( · ) 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 sizesn > 0.

Random variate generation

Further information:Non-uniform random variate generation

A conceptually very simple method for generating exponentialvariates is based oninverse transform sampling: Given a random variateU drawn from theuniform distribution on the unit interval(0, 1), the variate

$T=F^{-1}(U)$

has an exponential distribution, whereF⁻¹ is thequantile function, defined by

$F^{-1}(p)={\frac {-\ln(1-p)}{\lambda }}.$

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

$T={\frac {-\ln(U)}{\lambda }}.$

Other methods for generating exponential variates are discussed by Knuth^[24] and Devroye.^[25]

A fast method for generating a set of ready-ordered exponential variates without using a sorting routine is also available.^[25]

References

^"7.2: Exponential Distribution".Statistics LibreTexts. 2021-07-15. Retrieved2024-10-11.
^"Exponential distribution | mathematics | Britannica".www.britannica.com. Retrieved2024-10-11.
^^a ^bWeisstein, Eric W."Exponential Distribution".mathworld.wolfram.com. Retrieved2024-10-11.
^^a ^bNorton, Matthew; Khokhlov, Valentyn; Uryasev, Stan (2019)."Calculating CVaR and bPOE for common probability distributions with application to portfolio optimization and density estimation"(PDF).Annals of Operations Research.299 (1–2). Springer:1281–1315.arXiv:1811.11301.doi:10.1007/s10479-019-03373-1. Archived fromthe original(PDF) on 2023-03-31. Retrieved2023-02-27.
^Park, Sung Y.; Bera, Anil K. (2009)."Maximum entropy autoregressive conditional heteroskedasticity model"(PDF).Journal of Econometrics.150 (2). Elsevier:219–230.doi:10.1016/j.jeconom.2008.12.014. Archived fromthe original(PDF) on 2016-03-07. Retrieved2011-06-02.
^Michael, Lugo."The expectation of the maximum of exponentials"(PDF). Archived fromthe original(PDF) on 20 December 2016. Retrieved13 December 2016.
^Eckford, Andrew W.; Thomas, Peter J. (2016). "Entropy of the sum of two independent, non-identically-distributed exponential random variables".arXiv:1609.02911 [cs.IT].
^^a ^b ^c ^d ^e ^f ^g ^h ⁱLeemis, Lawrence M.; McQuestion, Jacquelyn T. (February 2008)."Univariate Distribution Relationships"(PDF).The American Statistician.62 (1): 45-53.doi:10.1198/000313008X270448.
^Ibe, Oliver C. (2014).Fundamentals of Applied Probability and Random Processes (2nd ed.). Academic Press. p. 128.ISBN 9780128010358.
^Richard Arnold Johnson; Dean W. Wichern (2007).Applied Multivariate Statistical Analysis. Pearson Prentice Hall.ISBN 978-0-13-187715-3. Retrieved10 August 2012.
^NIST/SEMATECH e-Handbook of Statistical Methods
^Elfessi, Abdulaziz; Reineke, David M. (2001)."A Bayesian Look at Classical Estimation: The Exponential Distribution".Journal of Statistics Education.9 (1).doi:10.1080/10691898.2001.11910648.
^Ross, Sheldon M. (2009).Introduction to probability and statistics for engineers and scientists (4th ed.). Associated Press. p. 267.ISBN 978-0-12-370483-2.
^Guerriero, V. (2012)."Power Law Distribution: Method of Multi-scale Inferential Statistics".Journal of Modern Mathematics Frontier.1:21–28.
^Severini, T. A. (2002-12-01)."On an exact probability matching property of right-invariant priors".Biometrika.89 (4):952–957.doi:10.1093/biomet/89.4.952.ISSN 0006-3444.
^Gerrard, R.; Tsanakas, A. (2011)."Failure Probability Under Parameter Uncertainty".Risk Analysis.31 (5):727–744.Bibcode:2011RiskA..31..727G.doi:10.1111/j.1539-6924.2010.01549.x.ISSN 1539-6924.PMID 21175720.
^Jewson, Stephen; Sweeting, Trevor; Jewson, Lynne (2025-02-20)."Reducing reliability bias in assessments of extreme weather risk using calibrating priors".Advances in Statistical Climatology, Meteorology and Oceanography.11 (1):1–22.Bibcode:2025ASCMO..11....1J.doi:10.5194/ascmo-11-1-2025.ISSN 2364-3579.
^Ritzema, H.P., ed. (1994).Frequency and Regression Analysis. Chapter 6 in: Drainage Principles and Applications, Publication 16, International Institute for Land Reclamation and Improvement (ILRI), Wageningen, The Netherlands. pp. 175–224.ISBN 90-70754-33-9.
^Lawless, J. F.; Fredette, M. (2005). "Frequentist predictions intervals and predictive distributions".Biometrika.92 (3):529–542.doi:10.1093/biomet/92.3.529.
^Bjornstad, J.F. (1990)."Predictive Likelihood: A Review".Statist. Sci.5 (2):242–254.doi:10.1214/ss/1177012175.
^Severini, Thomas A.; Mukerjee, Rahul; Ghosh, Malay (2002-12-01)."On an exact probability matching property of right-invariant priors".Biometrika.89 (4):952–957.doi:10.1093/biomet/89.4.952.ISSN 0006-3444.
^Jewson, Stephen; Sweeting, Trevor; Jewson, Lynne (2025-02-20)."Reducing reliability bias in assessments of extreme weather risk using calibrating priors".Advances in Statistical Climatology, Meteorology and Oceanography.11 (1):1–22.Bibcode:2025ASCMO..11....1J.doi:10.5194/ascmo-11-1-2025.ISSN 2364-3579.
^D. F. Schmidt and E. Makalic, "Universal Models for the Exponential Distribution",IEEE Transactions on Information Theory, Volume 55, Number 7, pp. 3087–3090, 2009doi:10.1109/TIT.2009.2018331
^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.
^^a ^bLuc Devroye (1986).Non-Uniform Random Variate Generation. New York: Springer-Verlag.ISBN 0-387-96305-7.Seechapter IX, section 2, pp. 392–401.

External links

The WikibookProbability has a page on the topic of:Exponential distribution

Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Power function Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling'sT-squared Hartman–Watson Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher'sz Kaniadakis κ-Gaussian Gaussianq Generalized hyperbolic Generalized logistic (logistic-beta) Generalized normal Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson'sS_U Landau Laplace Asymmetric Logistic Noncentralt Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student'st Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakisκ-exponential Kaniadakisκ-Gamma Kaniadakisκ-Weibull Kaniadakisκ-Logistic Kaniadakisκ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda