# Showing that Y has a uniform distribution if Y=F(X) where F is the cdf of continuous X

Let $$XX$$ be a random variable with a continuous and strictly increasing c.d.f. $$FF$$ (so that the quantile function $$F−1F^{−1}$$ is well-deﬁned). Deﬁne a new random variable $$YY$$ by $$Y=F(X)Y = F(X)$$. Show that $$YY$$ follows a uniform distribution on the interval $$[0,1][0, 1]$$.

My initial thought is that $$YY$$ is distributed on the interval $$[0,1][0,1]$$ because this is the range of $$FF$$. But how do you show that it is uniform?

Let $F_Y(y)$ be the CDF of $Y = F(X)$. Then, for any $y \in [0,1]$ we have:
$F_Y(y) = \Pr[Y \le y] = \Pr[F(X) \le y] = \Pr[X \le F^{-1}(y)] = F(F^{-1}(y)) = y$.