Let X be a random variable with a continuous and strictly increasing c.d.f. F (so that the quantile function F−1 is well-deﬁned). Deﬁne a new random variable Y by Y=F(X). Show that Y follows a uniform distribution on the interval [0,1].
My initial thought is that Y is distributed on the interval [0,1] because this is the range of F. But how do you show that it is uniform?
Let FY(y) be the CDF of Y=F(X). Then, for any y∈[0,1] we have:
What distribution has this CDF?