(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 )



, (6.65)

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 (6.30).

[(i): The probability in question is  ...  f1(z1)... fN (zN ) dz1 ··· dzN integrated over the set in which zi is the ti th smallest of the z’s fori = 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 <...