Editing Gumbel distribution (section)

==Related distributions==
===The discrete Gumbel distribution===
Many problems in [[discrete mathematics]] involve the study of an extremal parameter that follows a discrete version of the Gumbel distribution.<ref name=AguechAlthagafiBanderier>
{{Citation |arxiv=2311.13124|title=Height of walks with resets, the Moran model, and the discrete Gumbel distribution  |year=2023|first1=R.|last1=Aguech|first2=A.|last2=Althagafi|first3=C.|last3=Banderier|journal=Séminaire Lotharingien de Combinatoire|volume=87B|issue=12|pages=1–37}}</ref><ref>''Analytic Combinatorics'', Flajolet and Sedgewick.</ref> This ''discrete'' version is the law of <math>Y = \lceil X \rceil</math>, where <math>X</math> follows the ''continuous'' Gumbel distribution <math>\mathrm{Gumbel}(\mu, \beta)</math>.
Accordingly, this gives <math>P(Y \leq h) = \exp(-\exp(-(h-\mu)/\beta))</math> for any <math>h \in \mathbb Z</math>.

Denoting <math>\mathrm{DGumbel}(\mu, \beta)</math> as the discrete version, one has <math>\lceil X \rceil \sim \mathrm{DGumbel}(\mu, \beta)</math> and <math>\lfloor X \rfloor \sim \mathrm{DGumbel}(\mu - 1, \beta)</math>.

There is no known closed form for the mean, variance (or higher-order moments) of the discrete Gumbel distribution, but it is easy to obtain high-precision numerical evaluations via rapidly converging infinite sums. For example, this yields <math>{\mathbb E}[\mathrm{DGumbel}(0,1)]=1.077240905953631072609...</math>, but it remains an open problem to find a closed form for this constant (it is plausible there is none).

Aguech, Althagafi, and Banderier<ref name=AguechAlthagafiBanderier/> provide various bounds linking the discrete and continuous versions of the Gumbel distribution and explicitly detail (using methods from [[Mellin transform]]) the oscillating phenomena that appear when one has a sequence of random variables <math>\lfloor Y_n - c \ln n \rfloor</math> converging to a discrete Gumbel distribution.

===Continuous distributions===
* If <math>X</math> has a Gumbel distribution, then the conditional distribution of <math>Y=-X</math> given that <math>Y</math> is positive, or equivalently given that <math>X</math> is negative, has a [[Gompertz distribution]]. The cdf <math>G</math> of <math>Y</math> is related to <math>F</math>, the cdf of <math>X</math>, by the formula <math>G(y) = P(Y \le y) = P(X \ge -y \mid X \le 0) = (F(0)-F(-y))/F(0)</math> for <math>y>0</math>. Consequently, the densities are related by <math>g(y) = f(-y)/F(0)</math>: the [[Gompertz function|Gompertz density]] is proportional to a reflected Gumbel density, restricted to the positive half-line.<ref>{{Cite journal |doi=10.1016/j.insmatheco.2006.07.003 |title=Rational reconstruction of frailty-based mortality models by a generalisation of Gompertz' law of mortality |year=2007 |last1=Willemse |first1=W.J. |last2=Kaas |first2=R. |journal=Insurance: Mathematics and Economics |volume=40 |issue=3 |pages=468 |url=https://www.dnb.nl/binaries/Working%20Paper%20135-2007_tcm46-146792.pdf |access-date=2019-09-24 |archive-date=2017-08-09 |archive-url=https://web.archive.org/web/20170809050854/https://www.dnb.nl/binaries/Working%20Paper%20135-2007_tcm46-146792.pdf |url-status=dead }}</ref>
* If <math>X\sim\mathrm{Exponential}(1)</math> is an [[Exponential distribution|exponentially distributed]] variable with mean 1, then <math>\mu -\beta\log(X)\sim\mathrm{Gumbel}(\mu,\beta)</math>.
* If <math>U\sim\mathrm{Uniform}(0,1)</math> is a [[Continuous uniform distribution|uniformly distributed]] variable on the unit interval, then <math> \mu -\beta\log(-\log(U))\sim\mathrm{Gumbel}(\mu,\beta)</math>.
* If <math>X \sim \mathrm{Gumbel}(\alpha_X, \beta) </math> and <math> Y \sim \mathrm{Gumbel}(\alpha_Y, \beta) </math> are independent, then <math> X-Y \sim \mathrm{Logistic}(\alpha_X-\alpha_Y,\beta) \,</math> (see [[Logistic distribution]]).
* Despite this, if <math>X, Y \sim \mathrm{Gumbel}(\alpha, \beta) </math> are independent, then <math>X+Y \nsim \mathrm{Logistic}(2 \alpha,\beta)</math>. This can easily be seen by noting that <math>\mathbb{E}(X+Y) = 2\alpha+2\beta\gamma \neq 2\alpha = \mathbb{E}\left(\mathrm{Logistic}(2 \alpha,\beta) \right) </math> (where <math>\gamma</math> is the Euler-Mascheroni constant). Instead, the distribution of linear combinations of independent Gumbel random variables can be approximated by GNIG and GIG distributions.<ref name="Marques">{{Cite journal | last1=Marques|first1 = F.|  last2=Coelho| first2=C.| last3=de Carvalho|first3=M.| title = On the distribution of linear combinations of independent Gumbel random variables | journal=Statistics and Computing|year=2015|volume=25 | issue=3 | pages=683‒701| doi=10.1007/s11222-014-9453-5 | s2cid=255067312 | url=https://www.maths.ed.ac.uk/~mdecarv/papers/marques2015.pdf}}</ref>

Theory related to the [[generalized multivariate log-gamma distribution]] provides a multivariate version of the Gumbel distribution.