In this problem prove a generalization of Theorem 6.5. Given a random variable X with CDF Fx(x), define

F-(u) = min {x}Fx(x) > u }.

This problem proves that for a continuo us uniform (0, 1) random variable U, X =

F-(U) has CDF Fx-(x) Fx(x).

(a) Show that when F(x) is a continuous, strictly increasing function (i.e., X is not mixed, Fx(x) has no jump discontinuities, and F(x) has no "flat"intervals (a, b) where Fx(x) = c for a < x < b), then F-(u) = F-1x (u) for 0 < u < (1.

(b) Show that if Fx(x) has jump at x = x0, then F~(u) = x0 for all u in the interval

Fx(x0-) < u Fx(x+0).

(c) Prove that x^ = F^(U) has CDF Fx^(x) = Fx(x).