Order Statistics

我做到個答案但要12:00先post到﹐另外那條未有頭緒﹐又無答案

2008-03-29 00:05:25 補充：
use the formula we get
g(y1)=n(1-y1)^(n-1)
g(yn)=n(yn)^(n-1)

g(y1yn)=n(n-1)(yn-y1)^(n-2)

Consider Beta distribution
E(y1)=n/(n+1)
E(yn)=1/(n+1)

E(y1yn)
=∫∫y1yn n(n-1)(yn-y1)^(n-2) dyndy1 [y1 to 1][0 to 1]
=∫∫y1 yn n d(yn-y1)^(n-1)dy1 [y1 to ][0 to 1]
=n∫ y1 [(1-y1)^(n-1)-∫(yn-y1)^(n-1) dyn dy1 [y1 to 1][0 to 1]
=n∫ y1 [(1-y1)^(n-1)-(1-y1)^(n)/n dy1 [0 to 1]
=1/(n+1)-1/[(n+1)(n+2)] use Beta distribution again

So Cov(y1,yn)
=E(y1yn)-E(y1)E(yn)
=1/(n+1)-1/[(n+1)(n+2)]-[n/(n+1)][1/(n+1)]
=1/(n+2)-[n/(n+1)^2]
=1/[(n+1)^2(n+2)]