Editing Range (statistics) (section)

==Range of continuous IID random variables==

For ''n'' [[independent and identically distributed random variables|independent and identically distributed continuous random variables]] ''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub> with the [[cumulative distribution function]]  G(''x'') and a [[probability density function]] g(''x''), let T denote the range of them, that is, T= max(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>)-
min(''X''<sub>1</sub>, ''X''<sub>2</sub>, ..., ''X''<sub>''n''</sub>).

===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" />

===Moments===

The mean range is given by<ref>{{cite journal | author1 = H. O. Hartley | author-link1 = H. O. Hartley | author2 = H. A. David | year = 1954 | title = Universal Bounds for Mean Range and Extreme Observation | journal = The Annals of Mathematical Statistics | volume = 25 | issue = 1 | pages = 85–99 | jstor = 2236514 | doi=10.1214/aoms/1177728848| doi-access = free }}</ref>
::<math>n \int_0^1 x(G)[G^{n-1}-(1-G)^{n-1}] \,\text{d}G</math>
where ''x''(''G'') is the inverse function. In the case where each of the ''X''<sub>''i''</sub> has a [[standard normal distribution]], the mean range is given by<ref>{{cite journal | author = L. H. C. Tippett | author-link = L. H. C. Tippett | year = 1925 | title = On the Extreme Individuals and the Range of Samples Taken from a Normal Population | journal = Biometrika | volume = 17 | issue = 3/4 | pages = 364–387 | jstor = 2332087 | doi=10.1093/biomet/17.3-4.364}}</ref>
::<math display="block">\int_{-\infty}^\infty (1-(1-\Phi(x))^n-\Phi(x)^n ) \,\text{d}x.</math>

===Derivation of the distribution===

Please note that the following is an informal derivation of the result. It is a bit loose with the calculation of the probabilities.

Let <math>m, M</math> denote respectively the min and max of the random variables <math>X_1 \dots X_n</math>.

The event that the range is smaller than <math>T</math> can be decomposed into smaller events according to:

* the index of the minimum value
* and the value <math>x</math> of the minimum.

For a given index <math>i</math> and minimum value <math>x</math>, the probability of the joint event:

# <math>X_i</math> is the minimum,
# and <math>X_i=x</math>,
# and the range is smaller than <math>T</math>,

is:<math display="block">
g(x) \left[ G(x+T) - G(x) \right]^{n-1}
</math>Summing over the indices and integrating over <math>x</math> yields the total probability of the event: "the range is smaller than <math>T</math>" which is exactly the cumulative density function of the range:<math display="block">
F(t) = n \int_{-\infty}^{\infty} g(x) \left[G(t+x)-G(x) \right]^{n-1} \, \text{d}x
</math>which concludes the proof.