1 Distribution of $F\left(x\right)$

Suppose a univariate continuous random variable $x$’s probability density function (PDF) is $f\left(x\right)$ and its cumulative distribution function (CDF) $F\left(x\right)$. $F\left(x\right)$ itself is also a random variable, so what is $F\left(x\right)$’s distribution?

$F\left(x\right)$ has a couple of properties.

• $F\left(x\right)$’s range is [0, 1].
• $F(\cdot )$ is a monotonically non-decreasing function, but all commonly seen $F(\cdot )$ monotonically increases. In this case, $F^{-1}(\cdot )$ exists and also monotonically increases.

Let $y=F\left(x\right)$. $y$’s CDF is

Take its derivative, we have $y=F\left(x\right)\sim U(0,1)$.

2 Uniform Distribution to Other Continuous Distributions

Suppose $x\sim U(0,1)$, what kind of transformation $G$ can be applied to $x$ such that $y=G\left(x\right)$ follows a given distribution with PDF $f\left(y\right)$ and CDF $F\left(y\right)$?

Section 1 explains the transformation of a random variable from a distribution to $U(0,1)$, it seems to suggest the inverse transformation might work for this problem. That is, $F^{-1}$ might be the $G$ needed. Lets’s do the math. $F^{-1}\left(x\right)$’s CDF is

so we are correct. $F^{-1}\left(x\right)$’s CDF is indeed $F\left(y\right)$ and its PDF $f\left(y\right)$.

3 One General Distribution to Another

Given a random variable $x$ with PDF $f\left(x\right)$ and CDF $F\left(x\right)$, how can we transform it to another random variable $y$ with PDF $g\left(y\right)$ and CDF $G\left(y\right)$?

Based on Section 1 and Section 2, $y=G^{-1}\left(F\right(x\left)\right)$ is what we are looking for. The proof is obvious. Figure 1 may help one visualize and memorize the transformation steps.