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.