Let $F: \mathbb R \rightarrow [0,1]$ be strictly monotonic increasing distribution function. The random variable $X$ has distribution function $F$ and the random variable $U$ is uniformly distributed on $[0,1]$. I want to determine the distributions of $F(X)$ and $F^{-1}(U)$, but don't know how to do that.
Can somebody help? Thanks in advance!
Let $Y = F(X)$. Then $$\begin{align} F_{Y}(y) = \mathbb{P}\left(Y \leq y\right) &= \mathbb{P}\left(F(X) \leq y\right) \\& = \mathbb{P}\left(X \leq F^{-1}(y)\right)\text{ since }F\text{ is strictly increasing, }F^{-1}\text{ exists} \\ &= F_{X}\left(F_{X}^{-1}(y)\right) \\ &= y\text{,}\qquad y \in [0, 1]\text{.} \end{align}$$ The solution for finding the distribution of $F^{-1}(U)$ is similar.
