33. Distribution of order statistics. (i) If Z., Z is a sample from a cumulative distribution function F with density

f, the joint density of Y, Z,), i = 1,..., n, is (62) N!f(y)... f(y) (1 1)(551)!...(N-s)! x[F()][F(2) = F()]..[1 F(y)] N- - for y < < Y- (ii) For the particular case that the Z's are a sample from the uniform distribution on (0, 1), this reduces to N! (63) (5 1)(2 s 1)!...(Ns)! - - - ()(1). which For n 1, (63) is the density of the beta-distribution BN- therefore is the distribution of the single order statistic Z,, from U(0, 1).

(iii) Let the distribution of Y,..., Y, be given by (63), and let V, be defined by Y, VVV for i = 1,..., n. Then the joint distribution of the V, is N! - (5-1)! (N s)! (1-0)--- (+N+1) so that the V, are independently distributed according to the beta-distri- bution B-" [(i): If YZ Y = Z) and Y+ Y are the remaining Z's in the original order of their subscripts, the joint density of Y,..., Y is N(N- 1)...(N n+1)... ff(y+1)... f(y) dy+1. dyy integrated over the re- gion in which s-1 of the y's are