Editing Exponential distribution (section)

==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]].