Editing Order statistic (section)

==== The joint distribution of the order statistics of the uniform distribution ====

Similarly, for ''i''&nbsp;<&nbsp;''j'', the [[joint probability distribution|joint probability density function]] of the two order statistics ''U''<sub>(''i'')</sub>&nbsp;<&nbsp;''U''<sub>(''j'')</sub> can be shown to be

:<math>f_{U_{(i)},U_{(j)}}(u,v) = n!{u^{i-1}\over (i-1)!}{(v-u)^{j-i-1}\over(j-i-1)!}{(1-v)^{n-j}\over (n-j)!}</math>

which is (up to terms of higher order than <math>O(du\,dv)</math>) the probability that ''i''&nbsp;−&nbsp;1, 1, ''j''&nbsp;−&nbsp;1&nbsp;−&nbsp;''i'', 1 and ''n''&nbsp;−&nbsp;''j'' sample elements fall in the intervals <math>(0,u)</math>, <math>(u,u+du)</math>, <math>(u+du,v)</math>, <math>(v,v+dv)</math>, <math>(v+dv,1)</math> respectively.

One reasons in an entirely analogous way to derive the higher-order joint distributions. Perhaps surprisingly, the joint density of the ''n'' order statistics turns out to be ''constant'':

:<math>f_{U_{(1)},U_{(2)},\ldots,U_{(n)}}(u_{1},u_{2},\ldots,u_{n}) = n!.</math>

One way to understand this is that the unordered sample does have constant density equal to 1, and that there are ''n''! different permutations of the sample corresponding to the same sequence of order statistics. This is related to the fact that 1/''n''! is the volume of the region <math>0<u_1<\cdots<u_n<1</math>. It is also related with another particularity of  order statistics of uniform random variables: It follows from the [[BRS-inequality]] that the maximum expected number of uniform U(0,1] random variables one can choose from a sample of size n with a sum up not exceeding <math>0 <s <n/2</math> is bounded above by
<math> \sqrt{2sn} </math>, which is thus invariant on the set of all <math> s, n </math>
with constant product <math> s n </math>.

Using the above formulas, one can derive the distribution of the range of the order statistics, that is the distribution of <math>U_{(n)}-U_{(1)}</math>, i.e. maximum minus the minimum. More generally, for <math>n\geq k>j\geq 1</math>, <math>U_{(k)}-U_{(j)} </math> also has a beta distribution: <math display="block">U_{(k)}-U_{(j)}\sim \operatorname{Beta}(k-j, n-(k-j)+1)</math>From these formulas we can derive the covariance between two order statistics:<math display="block">\operatorname{Cov}(U_{(k)},U_{(j)})=\frac{j(n-k+1)}{(n+1)^2(n+2)}</math>The formula follows from noting that <math display="block">\operatorname{Var}(U_{(k)}-U_{(j)})=\operatorname{Var}(U_{(k)}) + \operatorname{Var}(U_{(j)})-2\cdot \operatorname{Cov}(U_{(k)},U_{(j)})
=\frac{k(n-k+1)}{(n+1)^2(n+2)}+\frac{j(n-j+1)}{(n+1)^2(n+2)}-2\cdot \operatorname{Cov}(U_{(k)},U_{(j)})</math>and comparing that with <math display="block">\operatorname{Var}(U)=\frac{(k-j)(n-(k-j)+1)}{(n+1)^2(n+2)}</math>where <math>U\sim \operatorname{Beta}(k-j,n-(k-j)+1)</math>, which is the actual distribution of the difference.