31. Let Z, have a continuous cumulative distribution function F, (i = 1,..., N), and let G be the group of all transformations Z-f(Z) such that fis continuous and strictly increasing. (i) The transformation induced by f in the space of distributions is F = F,(). (ii) Two N-tuples of distributions (E, F) and (F... F) belong to the same orbit with respect to G if and only if there exist continuous distribution functions h...., hy defined on (0,1) and strictly increasing continuous distribution functions F and F such that F = h, (F) and F-h,(F). [(i): P{f(Z) sy) P(Z, sf'(y)} = F[f'(y)]. (ii): If Fh,(F) and the F are on the same orbit, so that F' = F(), then Fh,(F) with F = F(f). Conversely, if Fh, (F), F = h,(F), then FF

(f) with f = F(F).]