Question: For a c.d.f. F with quantile function defined by F1 (u) = inf{x : F(x) u} , show that: (i) F(x) u is

For a c.d.f. F with quantile function defined by F−1

(u) = inf{x : F(x) ≥ u} , show that: (i) F(x) ≥ u is equivalent to F−1(u) ≤ x.

(ii) F−1(·) is nondecreasing and left continuous with right-hand limits.

(iii) F(F−1(u)) ≤ u with equality if F is continuous at F−1(u).

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