Let F be an arbitrary c.d.f. (not necessarily discrete, not necessarily continuous, not necessarily either). Let F−1 be the quantile function from Definition 3.3.2. Let X have the uniform distribution on the interval [0, 1]. Define Y = F−1(X). Prove that the c.d.f. of Y is F. Compute Pr(Y ≤ y) in two cases. First, do the case in which y is the unique value of x such that F(x) = F(y). Second, do the case in which there is an entire interval of x values such that F(x) = F(y).