Let F (x) be a cumulative distribution function and let
F−1 (u) = inf {x : F (u) x} .
Notice that if F (x) is continuous and strictly increasing then F−1 (u) is the
ordinary inverse function of F (x) .
(a) Show that
u : F−1 (u) x
= {u : u F (x)}
(b) Use (a) to prove that
F−1 (U) s F.
Briefly explain how this result can be used in practice.
(c) Suppose now that F (x) is continuous. Show that
F (X) s Unif (0, 1)
Hint: for part (c) you can assume that F (x) is strictly increasing (to facilitate
the proof) but this condition is not actually needed.