Editing Range (statistics) (section)

===Distribution===

The range, T,  has the cumulative distribution function<ref name="gumbel">{{cite journal | author = E. J. Gumbel | author-link = E. J. Gumbel | year = 1947 | title = The Distribution of the Range | journal = The Annals of Mathematical Statistics | volume = 18 | issue = 3 | pages = 384–412 | jstor = 2235736 | doi=10.1214/aoms/1177730387| doi-access = free }}</ref><ref name="tsimashenka">{{Cite book | last1 = Tsimashenka | first1 = I. | last2 = Knottenbelt | first2 = W. | last3 = Harrison | first3 = P. | author-link3 = Peter G. Harrison| doi = 10.1007/978-3-642-30782-9_12 | chapter = Controlling Variability in Split-Merge Systems | title = Analytical and Stochastic Modeling Techniques and Applications | series = Lecture Notes in Computer Science | volume = 7314 | pages = 165 | year = 2012 | isbn = 978-3-642-30781-2 | url = http://www.doc.ic.ac.uk/~wjk/publications/tsimashenka-knottenbelt-harrison-asmta-2012.pdf}}</ref>
::<math>F(t)= n \int_{-\infty}^\infty g(x)[G(x+t)-G(x)]^{n-1} \, \text{d}x.</math>
[[Emil Julius Gumbel|Gumbel]] notes that the "beauty of this formula is completely marred by the facts that, in general, we cannot express ''G''(''x''&nbsp;+&nbsp;''t'') by ''G''(''x''), and that the numerical integration is lengthy and tiresome."{{R|gumbel|p=385 <!-- (PDF p.&nbsp;2) -->}}

If the distribution of each ''X''<sub>''i''</sub> is limited to the right (or left) then the asymptotic distribution of the range is equal to the asymptotic distribution of the largest (smallest) value. For more general distributions the asymptotic distribution can be expressed as a [[Bessel function]].<ref name="gumbel" />