(i) Let Z1,...,ZN be independently distributed with densities f1,...,fN , and let the rank of Zi be denoted by Ti. If f is any probability density which is positive whenever at least one of the fi is positive, then P{T1 = t1,...,TN = tn} = 1 N!

E



f1



V(t1)



f



V(t1)

 ··· fN



V(tN )



f



V(tN )





. (6.62)

where V(1) < ··· < V(N) is an ordered sample from a distribution with density f.

(ii) If N = m + n, f1 = ··· = fm =

f, fm+1 = ··· = fm+n = g, and S1 < ··· < Sn denote the ordered ranks of Zm+1,...,Zm+n among all the Z’s, the probability distribution of S1,...,Sn is given by (6.27).

[(i): The probability in question is ... f1(z1) ...fN (zN ) dz1 ··· dzN integrated over the set in which zi is the tith smallest of the z’s for i = 1,...,N. Under the transformation wti = zi the integral becomes ... f1(wt1 ) ...fN (wtN ) dw1 ··· dwN integrated over the set w1 < ··· < wN . The desired result now follows from the fact that the probability density of the order statistics V(1) < ··· < V(N) is N!f(w1) ··· f(wN ) for w1 <...