Editing Gumbel distribution (section)

===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]].