Distribution of $F^{-1}(U)$
If $U \sim \text{Uniform}(0,1)$ and $F$ is the CDF of a random variable $X$, what is the distribution of $F^{-1}(U)$?
Answer
If $U \sim \text{Uniform}(0,1)$, then
\[
X = F^{-1}(U) \sim F,
\]
i.e. $F^{-1}(U)$ has the same distribution as $X$.
Indeed, for any $x$,
\[
P(F^{-1}(U) \leq x) = P(U \leq F(x)) = F(x).
\]
Hence, $F^{-1}(U)$ follows the distribution with CDF $F$.
Notes and comments
Comment 1: For intuition, just reverse what you did (mentally) for the x-axis and y-axis in the previous example.
Comment 2: This is the basis of the inverse transform method used to generate random variables from any distribution.