Editing Order statistic (section)

=== Cumulative distribution function of order statistics ===

For a random sample as above, with cumulative distribution <math>F_X(x)</math>, the order statistics for that sample have cumulative distributions as follows<ref>{{cite book 
|last1=Casella 
|first1=George 
|last2=Berger 
|first2=Roger 
|title=Statistical Inference 
|year=2002 
|publisher=Cengage Learning 
|isbn=9788131503942 
|page=229 
|edition=2nd
|url={{Google books|0x_vAAAAMAAJ|page=228|plainurl=yes}}
}}</ref>
(where ''r'' specifies which order statistic):
<math display="block">
F_{X_{(r)}}(x) = \sum_{j=r}^{n} \binom{n}{j} [ F_{X}(x) ]^{j} [ 1 - F_{X}(x) ]^{n-j}
</math>
The proof of this formula is pure [[combinatorics]]: for the <math>r</math>th order statistic to be <math> \leq x </math>, the number of samples that are <math> > x </math> has to be between <math> 0 </math> and <math> n-r </math>. In the case that <math> X_{(j)} </math> is the largest order statistic <math> \leq x </math>, there has to be <math> j </math> samples <math> \leq x </math> (each with an independent probability of <math> F_X(x) </math>) and <math> n-j </math> samples <math> >x </math> (each with an independent probability of <math> 1 - F_X(x) </math>). Finally there are <math> \textstyle \binom{n}{j} </math> different ways of choosing which of the <math> n </math> samples are of the <math> \leq x </math> kind.

The corresponding probability density function may be derived from this result, and is found to be

:<math>f_{X_{(r)}}(x) = \frac{n!}{(r-1)!(n-r)!} f_{X}(x) [ F_{X}(x) ]^{r-1} [ 1 - F_{X}(x) ]^{n-r}.</math>

Moreover, there are two special cases, which have CDFs that are easy to compute.

:<math>F_{X_{(n)}}(x) = \operatorname{Prob}(\max\{\,X_1,\ldots,X_n\,\} \leq x)  = [ F_{X}(x) ]^n</math>

:<math>F_{X_{(1)}}(x) = \operatorname{Prob}(\min\{\,X_1,\ldots,X_n\,\} \leq x)  = 1- [ 1 - F_{X}(x) ]^n</math>

Which can be derived by careful consideration of probabilities.