(i) For any continuous cumulative distribution function F, define F −1(0) = −∞, F −1(y) = inf{x : F(x) = y} for 0

∞ if F(x) < 1 for all finite x, and otherwise inf{x : F(x)=1}. Then F[F −1(y)] = y for all 0 ≤ y ≤ 1, but F −1[F(y)] may be < y.

(ii) Let Z have a cumulative distribution function G(z) = h[F(z)], where F and h are continuous cumulative distribution functions, the latter defined over (0,1). If Y = F(Z), then P{Y

(iii) If Z has the continuous cumulative distribution function F, then F(Z) is uniformly distributed over (0, 1).

[(ii): P{F(Z) < y} = P{Z