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

(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 < F−1(y)} = F[F−1(y)] = y.]