Let $X$ be a continuous random variable with distribution function $F_{X}\left(x\right)$ and let $Y=F_{X}\left(x\right)$, show that $Y\sim U\left(0,1\right)$, where $U$ is the uniform distribution.
Since the density function $f_{X}\left(x\right)$ can be seen as the derivative of $F_{X}\left(x\right)$, this problem looks like an obvious application of the Change of Variable Theorem. Nevertheless, the statement of the theorem requires that the function in which the random variable is evaluated (composed) to be strictly monotonic (strictly increasing or strictly decreasing) and, as it is well known, distribution functions are non-decreasing only.
So my question is: can the Change of Variable Theorem somehow be applied in this case?
Thanks in advanced for any help provided.
Note: Sorry if there are any gramatical errors in the redaction, English is not my first language.
Best Answer
As Did wrote, let $Y=F_X(X)$ and note that $F_X$ need not be strictly increasing for the following arguments to holds. If we define the generalized inverse of $F_X$ by $F_X^{\leftarrow}(y) =\inf \{ x \in \mathbb{R}: F_X(x)\geq y \}$, then try to prove the following steps