Editing Order statistic (section)

==== Order statistics sampled from an exponential distribution ====

For <math>X_1, X_2, .., X_n</math> a random sample of size ''n'' from an [[exponential distribution]] with parameter ''λ'', the order statistics ''X''<sub>(''i'')</sub> for ''i'' = 1,2,3, ..., ''n'' each have distribution

::<math>X_{(i)} \stackrel{d}{=} \frac{1}{\lambda}\left( \sum_{j=1}^i \frac{Z_j}{n-j+1} \right)</math>

where the ''Z''<sub>''j''</sub> are iid standard exponential random variables (i.e. with rate parameter 1). This result was first published by [[Alfréd Rényi]].<ref>{{Citation
 | last1 = David
 | first1 = H. A.
 | last2 = Nagaraja
 | first2 = H. N.
 | title = Order Statistics
 | pages = 9
 | year = 2003
 | chapter = Chapter 2. Basic Distribution Theory
 | doi = 10.1002/0471722162.ch2 | series = Wiley Series in Probability and Statistics
 | isbn = 9780471722168
 }}</ref><ref>{{cite journal
 |last        = Rényi
 |first       = Alfréd | author-link = Alfréd Rényi
 |title       = On the theory of order statistics
 |journal     = [[Acta Mathematica Hungarica]]
 |volume      = 4
 |issue       = 3
 |pages       = 191–231
 |date        = 1953
 |doi         = 10.1007/BF02127580 | doi-access=free
}}</ref>