Editing Gumbel distribution (section)

==Occurrence and applications==
===Applications of the continous Gumbel distribution===
[[File:FitGumbelDistr.tif|thumb|320px|[[Distribution fitting]] with [[confidence band]] of a cumulative Gumbel distribution to maximum one-day October rainfalls.<ref>{{Cite web|url=https://www.waterlog.info/cumfreq.htm|title=CumFreq, distribution fitting of probability, free calculator|website=www.waterlog.info}}</ref> ]]

Gumbel has shown that the maximum value (or last [[order statistic]]) in a sample of [[random variable]]s following an [[exponential distribution]] minus the natural logarithm of the sample size <ref>{{Cite web|url=https://math.stackexchange.com/questions/3527556/gumbel-distribution-and-exponential-distribution?noredirect=1#comment7669633_3527556|title=Gumbel distribution and exponential distribution|website=Mathematics Stack Exchange}}</ref>  approaches the Gumbel distribution as the sample size increases.<ref>{{cite book |last=Gumbel |first= E.J. |year=1954 |asin=B0007DSHG4 |title=Statistical theory of extreme values and some practical applications |series=Applied Mathematics Series |volume= 33 |edition=1st |url= https://ntrl.ntis.gov/NTRL/dashboard/searchResults/titleDetail/PB175818.xhtml |publisher= U.S. Department of Commerce, National Bureau of Standards}}</ref>

Concretely, let <math> \rho(x)=e^{-x} </math> be the probability distribution of <math> x </math> and <math> Q(x)=1- e^{-x} </math> its cumulative distribution. Then the maximum value out of <math> N </math> realizations of <math> x </math> is smaller than <math> X </math> if and only if all realizations are smaller than <math> X </math>. So the cumulative distribution of the maximum value <math> \tilde{x} </math> satisfies
:<math>P(\tilde{x}-\log(N)\le X)=P(\tilde{x}\le X+\log(N))=[Q(X+\log(N))]^N=\left(1- \frac{e^{-X}}{N}\right)^N, </math>
and, for large <math> N </math>, the right-hand-side converges to <math> e^{-e^{(-X)}}. </math>

In [[hydrology]], therefore, the Gumbel distribution is used to analyze such variables as monthly and annual maximum values of daily rainfall and river discharge volumes,<ref name = "Oosterbaan">{{cite book |editor-last=Ritzema |editor-first=H.P. |first1=R.J. |last1=Oosterbaan |chapter=Chapter 6 Frequency and Regression Analysis |year=1994 |title=Drainage Principles and Applications, Publication 16 |publisher=International Institute for Land Reclamation and Improvement (ILRI) |location=Wageningen, The Netherlands |pages=[https://archive.org/details/drainageprincipl0000unse/page/175 175–224] |chapter-url=http://www.waterlog.info/pdf/freqtxt.pdf |isbn=90-70754-33-9 |url=https://archive.org/details/drainageprincipl0000unse/page/175 }}</ref> and also to describe droughts.<ref>{{cite journal |doi=10.1016/j.jhydrol.2010.04.035 |title=An extreme value analysis of UK drought and projections of change in the future |year=2010 |last1=Burke |first1=Eleanor J. |last2=Perry |first2=Richard H.J. |last3=Brown |first3=Simon J. |journal=Journal of Hydrology |volume=388 |issue=1–2 |pages=131–143 |bibcode=2010JHyd..388..131B}}</ref>
 
Gumbel has also shown that the [[estimator]] {{frac|''r''|(''n''+1)}} for the probability of an event &mdash; where ''r'' is the rank number of the observed value in the data series and ''n'' is the total number of observations &mdash; is an [[unbiased estimator]] of the [[cumulative probability]] around the [[Mode (statistics)|mode]] of the distribution. Therefore, this estimator is often used as a [[plotting position]].

===Occurrences of the discrete Gumbel distribution===
In [[combinatorics]], the discrete Gumbel distribution appears as a limiting distribution for the hitting time in the [[coupon collector's problem]]. This result was first established by [[Pierre-Simon de Laplace|Laplace]] in 1812 in his ''Théorie analytique des probabilités'', marking the first historical occurrence of what would later be called the Gumbel distribution.

In [[number theory]], the Gumbel distribution approximates the number of terms in a random [[partition of an integer]]<ref>{{cite journal |doi=10.1215/S0012-7094-41-00826-8 |title=The distribution of the number of summands in the partitions of a positive integer |year=1941 |last1=Erdös |first1=Paul |last2=Lehner |first2=Joseph |journal=Duke Mathematical Journal |volume=8 |issue=2 |pages=335}}</ref> as well as the trend-adjusted sizes of maximal [[prime gaps]] and maximal gaps between [[prime constellations]].<ref>{{cite journal |arxiv=1301.2242 |last=Kourbatov |first= A. |title=Maximal gaps between prime ''k''-tuples: a statistical approach |journal=Journal of Integer Sequences |volume=16 |year=2013|bibcode=2013arXiv1301.2242K }} Article 13.5.2.</ref>

In [[probability theory]], it appears as the distribution of the maximum height reached by discrete walks (on the lattice <math>{\mathbb N}^2</math>), where the process can be reset to its starting point at each step.<ref name=AguechAlthagafiBanderier/>

In [[analysis of algorithms]], it appears, for example, in the study of the maximum carry propagation in base-<math>b</math> addition algorithms.<ref>{{citation |title=The average time for carry propagation|year=1978|first1=Donald E.|last1=Knuth|journal=Nederlandse Akademie van Wetenschappen. Proceedings. Series A. Indagationes Mathematicae|volume=81|pages=238–242}}</ref>