In Theorem 2.1.10 the probability integral transform was proved, relating the uniform cdf to any continuous cdf. In this exercise we investigate the relationship between discrete random variables and uniform random variables. Let X be a discrete random variable with cdf Fx(x) and define the random variable Y as Y = Fx (X).
(a) Prove that Y is stochastically greater than a uniform(0,1); that is, if U ~ uniform (0,1), then
P(Y > y) ≥ P(U > y) = 1 -y, for all y, 0 < y < 1,
P(Y > y) ≥ P(U > y) = 1 - y, for some y, 0 < y < 1.
(Recall that stochastically greater was defined in Exercise 1.49.)
(b) Equivalently, show that the cdf of Y satisfies Fy(y) ≤ y for all 0 < y < 1 and FY(y) < y for some 0 < y < 1.