Exponential distribution

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>{{#invoke:citation/CS1|citation |CitationClass=web }}</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>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> In addition to being used for the analysis of Poisson point processes it is found in various other contexts.<ref name="Weisstein">{{#invoke:citation/CS1|citation |CitationClass=web }}</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, binomial, gamma, and 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 Template:Closed-open. If a random variable X has this distribution, we write Template:Math.

The exponential distribution exhibits 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 Template:Math, 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

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

File:Median exp.svg

The median is the preimage F⁻¹(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 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 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 moments 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 Template:Math 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 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

Tukey criteria for anomalies.Template:Citation needed

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 quartiles 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">Template:Cite journal</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)

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}} 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 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 Template:Closed-open 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>Template:Cite journal</ref>

Distribution of the minimum of exponential random variables

Let X₁, ..., X_n be independent exponentially distributed random variables with rate parameters λ₁, ..., λ_n. 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 X₁, ..., X_n do not all have parameter 0.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</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 statistics. 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 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>Template:Cite arXiv</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(μ, β⁻¹), then |X − μ| ~ Exp(β).<ref name="Leemis" />
• If X ~ U(0, 1) then −log(X) ~ Exp(1).
• If X ~ Pareto(1, λ), then log(X) ~ Exp(λ).<ref name="Leemis">Template:Cite journal</ref>
• If X ~ SkewLogistic(θ), then <math>\log\left(1 + e^{-X}\right) \sim \operatorname{Exp}(\theta)</math>.
• If X_i ~ 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_i ~ Exp(λ_i) then:
  • <math>kX \sim \operatorname{Exp}\left(\frac{\lambda}{k}\right)</math>, closure under scaling by a positive factor.
  • 1 + X ~ BenktanderWeibull(λ, 1), which reduces to a truncated exponential distribution.
  • ke^X ~ Pareto(k, λ).<ref name="Leemis" />
  • e^−λX ~ U(0, 1).
  • e^−X ~ Beta(λ, 1).<ref name="Leemis" />
  • Template:Sfrace^X ~ 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" />
  • Template:Nowrap.
  • <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(k, θ) (shape, scale parametrisation) then the marginal distribution of X is Lomax(k, 1/θ), the gamma mixture
  • λ₁X₁ − λ₂Y₂ ~ Laplace(0, 1).
  • min{X₁, ..., X_n} ~ Exp(λ₁ + ... + λ_n).
  • If also λ_i = λ then:
    • <math>X_1 + \cdots + X_k = \sum_i X_i \sim</math> Erlang(k, λ) = Gamma(k, λ) with integer shape parameter k and rate parameter λ.<ref>Template:Cite book</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_i − X_j ~ Laplace(0, λ⁻¹).
  • If also X_i are independent, then:
    • <math>\frac{X_i}{X_i + X_j}</math> ~ 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) ~ 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 Template:Nowrap; i.e., X has a chi-squared distribution with 2 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(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 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 = (x₁, ..., x_n) 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 Template:Em an unbiased estimator of <math>\lambda,</math> although <math>\overline{x}</math> Template:Em an unbiased<ref name="Dean W. Wichern-2007">Template:Cite book</ref> MLE<ref>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 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>Template:Cite journal</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>Template:Cite book</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 Template:Math is the Template:Math percentile of the chi squared distribution with v 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 Template:Math 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">Template:Cite journal</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 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 trials 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 decays, 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 mutations on a DNA strand, or between roadkills 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 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.

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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.

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>Template:Cite book</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 positions as part of the cumulative frequency analysis.

In operating-rooms management, the distribution of surgery duration for a category of surgeries with 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_n+1, conditioned on the observed samples x = (x₁, ..., x_n) 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

1. a frequentist confidence distribution, obtained from the distribution of the pivotal quantity <math>{x_{n+1}}/{\overline{x}}</math>;<ref>Template:Cite journal</ref>
2. a profile predictive likelihood, obtained by eliminating the parameter λ from the joint likelihood of x_n+1 and λ by maximization;<ref>Template:Cite journal</ref>
3. an objective Bayesian predictive posterior distribution, obtained using the non-informative Jeffreys prior 1/λ;
4. the Conditional Normalized Maximum Likelihood (CNML) predictive distribution, from information theoretic considerations.<ref>D. F. Schmidt and E. Makalic, "Universal Models for the Exponential Distribution", IEEE Transactions on Information Theory, Volume 55, Number 7, pp. 3087–3090, 2009 {{#invoke:doi|main}}</ref>

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 sample x. The Kullback–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 parameter λ₀ 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 Template:Nowrap, and Template:Nowrap 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 Template:Nowrap.

Random variate generation

A conceptually very simple method for generating exponential variates is based on inverse transform sampling: Given a random variate U drawn from the uniform distribution on the unit interval Template:Open-open, the variate

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

has an exponential distribution, where FTemplate:I sup 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. Template:ISBN. See section 3.4.1, p. 133.</ref> and Devroye.<ref name="Luc Devroye">Luc Devroye (1986). Non-Uniform Random Variate Generation. New York: Springer-Verlag. Template:ISBN. See 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

External links

• Template:Springer
• Online calculator of Exponential Distribution